Efficient Implementation of the hp-boundary Element Method for the Navier-Lamé Equation

Transcription

1 Fakultät für Mathematik und Wirtschaftswissenschaften Institut für Numerische Mathematik Masterareit Efficient Implementation of the hp-boundary Element Method for the Navier-Lamé Equation vorgelegt von Andreas Bantle am 9. August Gutachter. Prof. Dr. S. Funken. Prof. Dr. K. Uran

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3 Acknowledgements. First and foremost, I would like to express my gratitude to my advisor, Prof. S. A. Funken, who always offered me advice and support throughout my research. Without his expertise and excellent supervision this thesis would not have een possile. I would also like to extend my gratitude to Prof. K. Uran for serving on my thesis committee. Special thanks go to Markus Bantle for giving me advice, whenever I needed help, answering my questions very patiently and editing my thesis. I would also like to thank Natalie Selinski for editing my thesis. Last ut not least, I would like to thank my family and Franziska Merkle for supporting me during my master studies at University of Ulm. Ulm, August. i

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5 Contents Introduction The Navier-Lamé Equation 5. Theoretical Background 5. Derivation of the Navier-Lamé Equation and the Weak Formulation 7 3 The hp-bem for the Navier-Lamé Equation 3. Mixed Traction and Displacement Prolem 3. The Dirichlet Prolem and Symm s Integral Equation The Neumann Prolem and the Hypersingular Integral Equation 4 Associated Legendre Functions and Related Functions 5 4. Associated Legendre Functions and Loatto Shape Functions 5 4. Special Integrals Related to the Associated Legendre Functions 9 5 Realization of the hp-bem for the Navier-Lamé Equation 4 5. Geometrical Basics 4 5. Basis Functions of the Discrete Function Spaces Computation of the Galerkin Matrices The Single Layer Operator The Doule Layer Operator The Hypersingular Operator A Posteriori Error Estimators The Dirichlet Prolem and Symm s Integral Equation The Neumann Prolem and the Hypersingular Integral Equation The Mixed Prolem 59 6 Implementation 6 6. Overview on the epsbem Package 6 6. Integrals of the Associated Legendre Functions Implementation of the hp-bem Implementation of the Galerkin Matrices Mixed Prolem with Gliding Conditions 73 7 Numerical Results The Dirichlet Prolem The Neumann Prolem The Mixed Traction and Displacement Prolem 9 8 Conclusion 97

6 Contents A Calculation of the Integral O j,k B Matla Programs for the Assemly of the Galerkin Matrices v ix C Matla Programs for Solving Prolems with Gliding Conditions xxi D Content of the CD xxiii iv

7 Chapter Introduction With the technical development of computers, the influence of mathematical modeling and simulation on research is increasing significantly. Instead of developing expensive and time consuming prototypes, mathematical models are derived and simulated on computers. Most of the physical prolems can e descried y mathematical models using partial differential equations PDEs or integral equations IEs. In particular, prolems in the field of linear elasticity can e modeled y an elliptic PDE, namely the Navier-Lamé equation. A common example in linear elasticity that can e modeled with the Navier-Lamé equation is the Cook s memrane example. Cook s memrane, which is illustrated in Figure., is fixed on the left-hand side and exposed to a traction on the right-hand side. Fig..: Domain for the Cook s memrane example left with expected displacement right. In most cases the solution of the PDE cannot e calculated analytically and thus numerical methods are needed to approximate the solution. There are a variety of methods for approximating solutions to PDEs including the Finite Element Method FEM, the Finite Difference Method FDM and the Boundary Element Method BEM. All methods have oth advantages and disadvantages and thus a method is chosen ased on the need of the application. One of the main uses of the BEM is to PDE models in linear elasticity, such as the Navier-Lamé equation. The BEM differs significantly from the other methods, since the main idea of the BEM is to transform the PDE into an equivalent oundary IE instead of discretizing the PDE itself. Thus, only the oundary of the domain has to e discretized. This can reduce the computational effort and consequentially the computational time, particularly with complicated domains. Moreover, prolems on unounded domains can e solved easily with the BEM, whereas several prolems occur with the FEM or the FDM. However, for transforming the PDE in an equivalent integral equation

8 Chapter : Introduction an analytical representation of a fundamental solution is required, ut such a representation is not always provided. The BEM is a Galerkin method, meaning the oundary IE is multiplied with test functions in a weak sense leading to the variational formulation. Discretizing this variational formulation yields a system of linear equations with a large numer of degrees of freedom. The main computational effort arises from assemling this system of linear equations. Particularly for practical prolems, the accuracy and the reliaility of the results are of fundamental importance for the calculations. In order to improve the accuracy of the numerical results there exist two main approaches. One is to refine the discretization of the oundary, which leads to an h-method. The other is to increase the polynomial degree of the numerical solution, which leads to a p-method. By comining oth approaches cleverly we otain an hp-method, where the advantages of oth methods can e exploited. In general, the hp-method can achieve exponential convergence rates for the error, whereas the h- and the p-method only lead to algeraic convergence rates for the error. Figure. illustrates the estimated error for the Cook s memrane example. error uniform h-method uniform p-method adaptive hp-method 6 3 degrees of freedom Fig..: Estimated error for the Cook s memrane example using the different approaches for reducing the error. We see that with the exponential convergence rate of the hp-method accurate results can e achieved with a small numer of degrees of freedom. On most computers, a similar accuracy cannot e achieved with the h- or the p-method due to the limited memory or at least long computational times are needed. Goal of this Thesis The goal of this thesis is to develop an efficient implementation of the hp-bem for the Navier-Lamé equation, so that practical, two-dimensional prolems for linear

9 3 elasticity can e solved. For this purpose, we want to reduce the error close to machine precision harnessing the exponential convergence rate of the hp-method. Therefore, we derive analytical formulas for the assemly of the Galerkin matrix that can e staly and efficiently calculated for high polynomial degrees. Moreover, we want to take advantage of progress in computer hardware in order to perform the calculations very efficiently. In particular, since most computers today are equipped with multi-core processors we want to parallelize the routines. Additionally, this thesis focuses on integrating all routines into the epsbem framework see [], a software package for solving the Laplace equation with the hp- BEM. Thus, the structure of the routines for solving the Laplace and the Navier- Lamé equations must e similar, so that the user only has to adjust the main routines for solving the Navier-Lamé equation. For this purpose, the core routines have to e implemented in C, whereas the main routines have to e implemented in Matla. Outline This thesis is structured into of six main chapters. The first four chapters are devoted to the theory of the hp-bem for the Navier-Lamé equation, and the implementation and the numerical results are presented in the last two chapters. Chapter gives an introduction to linear elasticity. We egin y giving the analytical ackground and introducing some asic notations. Susequently, we derive the Navier-Lamé equation and its weak formulation y referencing to a model example. An introduction to the theory of the hp-bem for the Navier-Lamé equation is given in Chapter 3. In particular, we introduce the oundary integral operators and derive the system of linear equations for different types of oundary conditions, namely mixed, Dirichlet and Neumann oundary conditions. In Chapter 4 we explore the associated Legendre functions and related functions. This chapter serves as a asis for developing an efficient implementation of the hp-bem. The approach for implementing the hp-bem, that is investigated in this thesis, is to take the Legendre polynomials and the Loatto shape functions as ansatz-functions. Using this approach, the analytical computation of the entries in the Galerkin matrix can e reduced to evaluating integrals that are related to the associated Legendre functions. Consequently, we investigate the efficient calculation of these integrals, which is the asic requirement for the efficient implementation of the hp-bem. In Chapter 5 we descrie the realization of the hp-bem using analytical formulas. We egin y introducing the asis for the ansatz- and test-space, which is the main point for implementing an efficient hp-bem, and y defining a parametrization of the oundary elements. The main part in this chapter is to derive analytical formulas for calculating the Galerkin matrices of the oundary integrals operators efficiently, using the special integrals related to the associated Legendre functions. This chapter closes y introducing some error estimators for the Dirichlet, the Neumann and the mixed prolems that can e used for creating adaptive algorithms. The implementation of the hp-version of the BEM is descried in Chapter 6. Since all routines for the implementation of the hp-bem have to e integrated into the epsbem framework, we first give an overview on this software package. Furthermore, we descrie all routines that are implemented within the scope of this thesis. In particular, we go into detail on the implementation of the functions introduced

10 4 Chapter : Introduction in Chapter 4 and on the routines to calculate the integral operators with the formulas derived in Chapter 5. Additionally, we discuss the implementation of gliding conditions, i.e. a special type of oundary conditions, so that we can solve a wider range of practical prolems in linear elasticity. In the last chapter, we present the numerical results. We consider several standard examples and verify the convergence rates and the efficiency and reliaility of the error estimator. This chapter closes with a discussion on practical examples of linear elasticity.

11 Chapter The Navier-Lamé Equation. Theoretical Background In this section, we give the theory of Soolev spaces and introduce some important notations, that we need throughout this thesis, where we are guided y the work of [6]. Note that we only give a summary of the theory without proving the results. Throughout this section let Ω R n n e a domain. Let Cc Ω := {u C Ω : u has compact support} and the Leesgue space L e defined as usual with the L scalar product f, g L Ω := fx gx dx and the induced norm L Ω. Definition.. We call u L Ω weakly differentiale if there exist functions g,..., g n L Ω so that Ω Ω ux φx dx = g j x φx dx x j Ω j =,.., n and φ C c Ω and we call g,..., g n T the weak derivative of u. If u is differentiale, the usual derivative and the weak derivative are the same a.e.. Therefore, we write u = g,..., g n T for the weak derivative. Definition.. Soolev spaces on domains We define the Soolev space H Ω y the Leesgue Space L Ω, H Ω := L Ω equipped with the usual L norm. The Soolev space H Ω is given y H Ω := { u L Ω : u is weakly differentiale, u L Ω }. A norm on H is defined y u H Ω := u L Ω + u L Ω. Moreover, we define the Soolev space with positive integer order k y H k Ω := { u L Ω : u is weakly differentiale, u H k Ω }.

12 6 Chapter : The Navier-Lamé Equation A norm on H k Ω can e defined y A semi-norm on H k Ω is given y u H k Ω := u L Ω + u H k Ω. u H k Ω := D k u L Ω, where D k denotes the k-th weak derivative. Lemma..3 The Soolev spaces H k Ω, k N, are Hilert spaces with continuous emedding H k Ω L Γ. With the Riesz representation theorem we can define the dual space of the Soolev spaces as extensions of L. Definition..4 The dual space of the Soolev space H k Ω is defined y the Soolev space H k Ω of negative integer order k equipped with the usual operator norm u H k Ω := sup v H k Ω u, v v H k Ω where the duality rackets, :=, H k Ω H k Ω are the extended L scalar product. We also need to define the Soolev spaces with non-integer order on the oundary Γ := Ω. Therefore, we introduce the Soolev-Sloodeckij semi-norm for s, y u s,γ := Γ Γ ux uy x y ds n+s x ds y., Definition..5 Soolev spaces on the oundary i Let s,. Then, the Soolev space H s Γ is defined y H s Γ := { u L Γ : u H s Γ < } with the norm u H s Γ = u L Γ + u s,γ. ii For Γ Γ and s, we define H s Γ := {u = ũ Γ : ũ H s Γ } H s Γ := {u = ũ Γ : ũ H s Γ, supp ũ Γ }.

13 Section.: Derivation of the Navier-Lamé Equation and the Weak Formulation 7 Lemma..6 i H Γ is a Hilert space with continuous emedding H Γ L Γ. ii H Γ is the trace space of H Ω, i.e. H Γ = {u = ũ Γ : ũ H Ω}. Again, the Riesz representation theorem provides a definition of the dual spaces of the Soolev spaces with non-integer order. Definition..7 For s, we denote the dual space of H s Γ y H s Γ equipped with the usual operator norm on H s Γ that is given y u H s Γ := sup v H s Γ u, v, v H s Γ where the duality rackets, :=, H s Γ H s Γ are the extended L scalar product. Finally, we introduce the piecewise Soolev spaces for a given partition T h of the oundary Γ, i.e. Γ l, Γ k T h : Γ l Γ k = δ l,k Γ l and Γ = Γ l T h Γ l. Definition..8 For s the piecewise Soolev space is defined y equipped with the norm H s pwγ := { ψ H min{,s} Γ : ψ Γl H s Γ l, Γ l T h } ψ H s pw Γ := Γ l T h ψ H s Γ l. To simplify the notation we write s instead of H s Γ, s instead of H s Γ and k instead of H k Ω. Moreover, we restrict to write, for the different duality products that we defined aove. Since all functions that occur within this thesis are two dimensional functions u : R R, we write u V meaning u i V i =, for all aove defined function spaces. The corresponding norms are given y u V = u V + u V.. Derivation of the Navier-Lamé Equation and the Weak Formulation In this section we give a short introduction the the Navier-Lamé equation. The most important application of the Navier-Lamé equation is to linear elasticity. Therefore,

14 8 Chapter : The Navier-Lamé Equation we consider the mixed traction and displacement prolem of homogeneous, isotropic and linearly elastic odies. For a given ody that is exposed to oth volume forces and surface forces we want to calculate the displacement and the strain of the displaced ody. Additionally, the ody is fixed at some parts of the surface, in other words we prescrie the displacement to e zero, and at other parts of the surface a traction is given. An example is illustrated Figure.. Fig..: Model of a ar that is fixed on left lack shaded side. The ar is exposed to volume forces such as gravity red arrow and the right lue shaded side of the ar is exposed to a traction lue arrow. In many applications it is a good approximation to map the three-dimensional model of the ody to a two-dimensional model. Especially, in the case of plain stain or plain stress reducing the dimension of the ody is reasonale. In this work, we consider the plain strain state and thus, we only deal with two-dimensional models. Let Ω R e a ounded domain with Lipschitz-oundary Γ := Ω descriing the homogeneous, isotropic and linearly elastic ody. Considering mixed traction and displacement prolems we divide the oundary Γ into the Dirichlet oundary Γ D and the Neumann oundary Γ N. On the Dirichlet oundary, which is a closed suset of Γ with Γ D >, the exact displacement u D : Γ D R, u D CΓ D, of the ody is given, whereas on the Neumann oundary Γ N := Γ \Γ D we prescrie the traction y a function g : Γ N R, g L Γ N. The volume forces that are applied to the ody can e descried y a function f : Ω R, f CΩ. The sought displacement field of the ody is given y u : Ω R, where we assume u C Ω CΩ. Besides the visile displacement of the ody, external forces also result in inner strain and stress. Therefore, we define the strain tensor σ and the stress tensor ε of the displaced ody. Assuming a homogeneous, isotropic and linearly elastic material and small deformations, the strain and the stress tensors are symmetric matrices, i.e. σ, ε Rsymm. Moreover, the strain tensor can e approximated y the symmetric part of the gradient of the displacement field u,

15 Section.: Derivation of the Navier-Lamé Equation and the Weak Formulation 9 which reads εu u + u T. In linear elasticity, strain and stress are linearly related, i.e. Hook s law holds: σu = µ εu + λ divu I, where I R denotes the identity matrix and λ, µ R are the Lamé constants. The Lamé constants are dependent on the parameters of material E, which is called the modulus of elasticity, and ν, which is called the Poisson s ratio. For homogeneous, isotropic and linearly elastic materials there holds E > and ν,. In the plain strain state, to which we refer in this work, the Lamé constants and the parameters of material are related as follows: λ = E ν + ν ν and µ = E + ν. This relationship and the ranges of ν and E imply that µ > and λ > 3 µ. In the state of equilirium all forces add up to zero. Thus, we otain f + div σ = σ n = g, where n denotes the outer unit normal vector. Putting all results together we can formulate the mixed traction and displacement prolem: For given volume forces f CΩ, traction g L Γ N and displacement u D CΓ D, find u C Ω CΩ such that div σu = f in Ω u = u D on Γ D. σu n = g on Γ N, with σu = µ εu + λ divu I. Since. does not always have a solution u C Ω CΩ we consider the weak formulation of the prolem, where the solution u is supposed to e in H Ω. To this end, we define two trace operators in order to e ale to formulate the oundary conditions for the weak formulation. Definition.. i The trace operator γ is defined y γ ux := lim u x a.e. on Γ. Ω x x Γ

16 Chapter : The Navier-Lamé Equation ii The co-normal derivative γ is given y γ ux := where n denotes the outer normal vector of x. lim σ u x n a.e. on Γ.3 Ω x x Γ A detailed analysis of the trace operators can e found in [6], e.g. it is shown that these operators are well defined and can e extended to H Ω: γ : H Ω H Γ, γ : H Ω H Γ. Now we can state the weak formulation of.: For given volume forces f H Ω, traction g H Γ N and displacement u D H Γ D, find u H Ω such that div σu = f in Ω γ u = u D on Γ D.4 γ u = g on Γ N. Note that.4 holds in a weak sense.

17 Chapter 3 The hp-bem for the Navier-Lamé Equation In this chapter we give an introduction to the hp-bem for the Navier-Lamé equation. As the asic theory is well known and a detailed analysis is given in [6], [4] and [5] we just give a summary of the main results. Besides the mixed traction and displacement prolem, that we investigate in the first section, we also consider two special cases in this chapter, namely the Dirichlet prolem and the Neumann prolem. 3. Mixed Traction and Displacement Prolem Throughout this section, we consider the weak formulation of the mixed traction and displacement prolem.4. We assume Ω R to e a ounded domain with Lipschitz oundary Γ := Ω. The oundary is divided into the Neumann and the Dirichlet oundaries, i.e. Γ = Γ D Γ N with Γ D Γ N = and Γ D >. Furthermore, we ignore volume forces, i.e. f. First, we state the representation formula, which is also called the Somigliana identity. Theorem 3.. Let u H Ω with divσu = in a weak sense. Then, there holds for all x Ω ux = Γ Ux, y γ uy ds y Γ [γ,y Ux, y] γ uy ds y, 3. where the Kelvin matrix Ux, y is the fundamental solution of.4 and the trace operator γ,y with respect to y is applied to the columns of U, i.e. γ,y [U x, y, U x, y] := [γ,y U x, y, γ,y U x, y]. The Kelvin-matrix has the following representation regarding the Lamé parameters λ, µ R Ux, y = λ + 3µ 4πµ λ + µ where I R denotes the identity matrix. log x y I λ + µ x yx y T λ + 3µ x y, 3. Additionally, it can e shown that the traction T x, y := γ,y Ux, y of the Kelvin matrix has the following representation formula µ x y T ny µ T x, y = I + πλ + µ x y πλ + µ + λ + µ x y T ny x yx y T, πλ + µ x y 4 x y T ty x y I I 3.3

18 Chapter 3: The hp-bem for the Navier-Lamé Equation where ny denotes the outer unit normal vector and ty denotes the unit tangential vector with respect to y Γ. Moreover, the cross product for matrices is defined as the cross product of the rows and the columns which yields I I =. The representation formula 3. motivates the definition of the following oundary integral operators. Definition 3.. Let x R \Γ, φ H Γ and ψ H Γ. Then, the single layer potential Ṽ is given y Ṽ φx := Γ Ux, y φy ds y 3.4 and the doule layer potential K reads Kψx := [γ,y Ux, y]ψy ds y. 3.5 Γ One can show that oth Ṽ : H Γ H Ω and K : H Γ H Ω are linear and ounded operators. Plugging in the single and the doule layer potential, the representation formula now reads ux = Ṽ γ ux Kγ ux, x Ω. 3.6 Thus, using 3.6 we can calculate the solution u for all x Ω if we know the Cauchy-data γ u, γ u on the oundary Γ. Since the Neumann data γ u is only given on the Neumann oundary and the Dirichlet data γ u is only given on the Dirichlet oundary, we have to compute the missing Cauchy-data. In the following we derive a system of oundary integral equations that we use to compute the missing data. We first estalish some analytical results. Lemma 3..3 Let Ṽ : H Γ H Ω e the single layer potential and K : H Γ H Ω e the doule layer potential. Moreover, let x Γ, α e the inner angle at x and Bx, ε e the ε all around x. Then, there holds i γ Ṽ : H Γ H Γ defines a linear and ounded operator. L Γ there holds γ Ṽ φx = Γ For φ Ux, y φy ds y. 3.7 ii γ Ṽ : H Γ H Γ defines a linear and ounded operator. For φ H Γ there holds γ Ṽ φx = lim [γ,x Ux, y] φy ds y + ε Γ α φx, 3.8 \Bx,ε π

19 Section 3.: Mixed Traction and Displacement Prolem 3 iii γ K : H Γ H Γ defines a linear and ounded operator. For ψ H Γ we have to following formula γ Kψx = + α ψx + lim [γ,y Ux, y] ψy ds y. 3.9 π ε Γ \Bx,ε iv γ K : H Γ H Γ defines a linear and ounded operator. Note that the integrals in ii and iii exist and that α = π holds if x Γ has a normal vector. Lemma 3..3 motivates the following definition. Definition 3..4 Let x Γ, φ H Γ and ψ H Γ. Then, we define the following integral operators: i The single layer operator V : H Γ H Γ is given y V φx := Ux, y φy ds y. 3. Γ ii The doule layer operator K : H Γ H Γ is given y Kψx := lim [γ,y Ux, y] ψy ds y. 3. ε Γ \Bx,ε iii The adjoint operator K : H Γ H Γ is given y K φx := lim [γ,x Ux, y] φy ds y. 3. ε Γ \Bx,ε iv The hypersingular operator W : H Γ H Γ is given y W ψx := γ Kψx. 3.3 The properties of the integral operators can e summarized y the following lemma. Lemma 3..5 i The operators V, K, K and W are linear and ounded. ii The single layer operator V is symmetric with respect to the duality product, φ, V φ = φ, V φ φ, φ H Γ.

20 4 Chapter 3: The hp-bem for the Navier-Lamé Equation iii With appropriate scaling of the domain Ω the single layer operator is H - elliptic, in other words β > and c > : φ, V β φ c φ φ H Γ, where V β is given y V β φx = U β x, y φy ds y, Γ U β x, y = λ + 3µ 4πµ λ + µ log βx y I λ + µ x yx y T λ + 3µ x y iv The hypersingular operator is H -semi-elliptic, i.e. c > :. W ψ, ψ c ψ ψ H Γ. v K is the adjoint of K with respect to the duality product, i.e K φ, ψ = φ, Kψ φ H Γ, ψ H Γ. 3.4 vi Let d ds denote the arc length derivative. Then, there holds ψ, ψ H Γ W ψ, ψ = V d ds ψ, d ds ψ 3.5 with and V φx = Γ U x, y φy ds y U µ λ + µ x, y = log x y I π λ + µ x yx yt x y. Note that we do not amplify the choice of the scaling factor β in iii, as the single layer operator is elliptic in the examples considered in the susequent chapters. We refer to [6] for a detailed description of how to construct an appropriate β. In order to get a method for computing the missing Cauchy data we apply the trace operators γ and γ to the representation formula 3.6 which yields for x Γ γ ux = Ux, y γ uy ds y + α γ ux lim [γ,y Ux, y] γ uy ds y π ε Γ \Bx,ε and Γ γ ux = lim [γ,x Ux, y] γ uy ds y + ε Γ α \Bx,ε π γ ux γ,x Kγ ux.

21 Section 3.: Mixed Traction and Displacement Prolem 5 Assuming that x has a normal vector, i.e. α = π, and plugging in the oundary integral operators we write shorter γ u = I K V γ u γ u W I K γ u This system is called the Caldéron-system and can e used to compute the missing Cauchy-data. We choose an aritrary extension u D H Γ and g H Γ of the given data u D H Γ D and g H Γ N. Then, the missing Cauchy-data is given y un g D := γ u γ u ud. g It can e proven that u N H Γ N and g D H Γ D. Thus, we can reformulate 3.6 to un A = g D A ud on Γ g N Γ D, 3.7 where A := K V W K denotes the Caldéron-projector. To proceed to the equivalent variational formulation we define V := H Γ N H Γ D and its dual space V := H Γ D H Γ N using the duality product φ D, ψ N, φ N, ψ D V V := ψ D, φ D ΓD + ψ N, φ N ΓN. Here,, ΓN and, ΓD denote the extended L scalar product on Γ N and Γ D, respectively. One can show that the Coldéron-projector defines a linear and ounded operator A : V V. Moreover, we define the continuous ilinear form a on V V y a u N, g D, φ N, ψ D := Au N, g D, φ N, ψ D V V. 3.8 The variational formulation that is equivalent to 3.7 reads: Find u N, g D V such that φ N, ψ D V there holds a u N, g D, φ N, ψ D = A u D, g, φ N, ψ D V V. 3.9 In order to e ale to make a statement concerning the existence and the uniqueness of a solution we cite the following lemma from [6]. Lemma 3..6 Lax-Milgram Let X e a Hilert space, X e the dual space of X and let A : X X e ounded and X-elliptic. Then, there holds f X! u X : Au = f.

22 6 Chapter 3: The hp-bem for the Navier-Lamé Equation Theorem 3..7 Existence of a unique solution The integral formulation of the mixed traction and displacement prolem 3.7 and the equivalent variational formulation 3.9, respectively, have an unique solution. Proof. Due to the previous results and the Lax-Milgram lemma it remains to show that the operator A is V-elliptic. Using the H -ellipticity of V, the H -semiellipticity of W and the continuity of oth operators one can show that u N, g D A := V g D, g D ΓD + W u N, u N ΓN defines a norm on V that is equivalent to the usual product norm V. Moreover, let u N, g D V, then there holds A u N, g D, u N, g D V V = Ku N + V g D, g D ΓD + W u N + K g D, u N ΓN which proofs the V-ellipticity of A. = V g D, g D ΓD + W u N, u N ΓN = u N, g D A c u N, g D V, In order to solve the variational formulation 3.9 numerically, we prescrie that Ω R is a polygonal Lipschitz domain with the oundary Γ := Ω. Note that if Ω is not a polygonal domain we have to approximate it y a polygonal domain which leads to an approximation error. However, this case is not considered in this work. Let T h e a triangulation of the oundary Γ with { } N el T h := Γ l = conv{a l, B l }, l =,..., N el : Γ = Γ l, Γ l Γ m = δ lm Γ l, where h := max{ Γ l : l N el }. Note that we call the elements of T h oundary elements. In order to e ale to solve the mixed prolem with different oundary conditions, we define the triangulations T h,d of the Dirichlet and T h,n of the Neumann oundaries, where we assume T h = T h,d T h,n. Note that this definition implies that for all Γ l T h we get either Γ l T h,d or Γ l T h,n. Furthermore, we denote y N el,d and N el,n the numer of oundary elements on the Dirichlet and the Neumann oundaries, respectively. In order to get a discrete prolem, we approximate the infinite-dimensional space V = H Γ N H Γ D y an appropriate finite-dimensional space V h := S T h,n, p S T h,d, p where p N N el,d, p N N el,n. S T h,n, p contains piecewise polynomial, gloally continuous functions, i.e. { S T h,n, p := ψ p h CΓ } N : ψ p h Γl P p,l, l =,..., N el,n. and S T h,d, p contains piecewise polynomial, gloally discontinuous functions, i.e. { S T h,d, p := φ p h } L Γ D : φ p h Γl P p,l, l =,..., N el,d. l=

23 Section 3.: Mixed Traction and Displacement Prolem 7 Note that we otain a conforming method for solving the discrete prolem, since there holds V h V. The discrete prolem reads Find u N,h, g D,h V h such that φ N,h, ψ D,h V h there holds a u N,h, g D,h, φ N,h, ψ D,h = A u D, g, φ N,h, ψ D,h V V. 3. To make a statement concerning the solvaility of the discrete prolem, we cite the following lemma from [6]. Lemma 3..8 Céa Let X e a Hilert space, a : X X R a continuous and X-elliptic ilinear form and f X a continuous linear form. Moreover, let X h X e a finite-dimensional test- and ansatz-space. formulation Then, the discrete variational a u h, v h = fv h v h X h has a unique solution u h X h and u h is the est approximation in the sense that u u h X c inf v h X h u v h X. In particular, the Galerkin orthogonality holds, i.e. u u h, v h X = v h X h. The fact that the Caldéron-projector A is V-elliptic and continuous implies the continuity and the ellipticity of the ilinear form a and thus, we can apply the Céa lemma to 3.. Hence, 3. is well defined and the solution u N,h, g D,h converges quasi-optimal to the exact solution. To solve 3. we choose a asis Θ j j=,...,dim Vh of V h and we write u N,h, g D,h = dim Vh k= α k Θ k, α k R. Testing 3. with the asis functions Θ j and plugging in the representation of u N,h, g D,h yields dim V h k= α k a Θ k, Θ j = A u D, g, Θ j V V j =,..., dim V h. 3. In order to write 3. shorter, we define the Galerkin matrix A := a j,k j,k=,...,dim Vh y a j,k := a Θ k, Θ j, the mass matrix M = m j,k j,k=,...,dim Vh y m j,k := Θ j, Θ k V V, the coefficient vector x := α k k=,...,dim Vh and the coefficient vector := j j=,...,dim Vh of u D, g. Finally, the discrete prolem is given y the system of linear equations A x = M A.

24 8 Chapter 3: The hp-bem for the Navier-Lamé Equation Descriing the Galerkin matrix more closely we denote y Φ j j=,...,dim S T h,p the asis of S T h, p and y Ψ j j=,...,dim S T h,p the asis of S T h, p. Then, the Galerkin matrix of the single layer operator is defined y V := V i,j i,j=,...,dim S T h,p with V i,j := V Φ j, Φ i, the Galerkin matrix of the doule layer operator is defined y K := K i,j j=,...,dim S T h,p i=,...,dim S T h,p with K i,j := KΨ j, Φ i and the Galerkin matrix of the hypersingular operator is given W := W i,j i,j=,...,dim S T h,p with W i,j := W Ψ j, Ψ i. Then, there holds K V A = W K T. Note that we used the fact that K the adjoint of K which yields that the Galerkin matrix of K is the transposed Galerkin matrix of the doule layer operator. In the end of this section we give an important result concerning the a priori error analysis for the mixed traction and displacement prolem. Theorem 3..9 Let ψ, φ V e the solution of 3.9. Moreover, let T h e a triangulation of Γ with h := max{ Γ l : Γ l T h } and ψ p h, φp h V h = S T h, p S T h, p denote the unique solution of the discrete prolem 3. with polynomial degrees p N and p N on the mesh T h. Prescriing the regularity of the exact solution ψ, φ H t pwγ N H s pwγ D for s, t, there holds ψ, φ ψ p h, φp h H Γ H Γ h min{t,p + } ψ H t pw Γ N + h min{s+,p + 3 } φ H s pw Γ D. 3. Proof. See [5], page shows that if the exact solution is smooth, the convergence rate of the error is only ounded y the polynomial degree of the discrete solution and if the exact solution is not smooth, e.g. in the neighorhood of a singularity, small polynomial degrees suffice to otain the maximal convergence rate. This motivates the implementation of an hp-method. The idea of the hp-method is that we increase the polynomial degree on the oundary elements, on which the exact solution is smooth, and refine the oundary mesh near a singularity of the exact solution in order to otain the maximal convergence rate. Therey, we implement oth geometric and adaptive hp-methods. Implementing a geometric hp-method, we know a priori where the singularity of the exact solution is and thus, we prescrie a geometric hp-refinement, i.e. we refine the oundary near the singularity and choose p = on the smallest element and increase the polynomial degree on the igger elements. Implementing an adaptive hp-method we control the hp-refinement y an appropriate error estimator. The numerical results in Chapter 7 show an exponential convergence rate for oth the geometric and the adaptive hp-methods. Note that the exponential convergence is not yet proven theoretically for adaptive hp-methods, whereas in [7] there is a proof for the geometric hp-method. In addition to the hp-method, we also implement adaptive h- and uniform h- and p-methods. Note that in [7] it is proven that the convergence rate of the uniform p-method is twice as ig compared to the convergence rate of the uniform h-method.

25 Section 3.: The Dirichlet Prolem and Symm s Integral Equation 9 3. The Dirichlet Prolem and Symm s Integral Equation In this section we investigate the Dirichlet prolem, which reads { div σu = in Ω γ u = u D on Γ, 3.3 where u D H Γ denotes the given Dirichlet data. In order to solve the Dirichlet prolem we compute the missing Neumann data φ H Γ using the first equation of the Caldéron-sytem, which is called Symm s integral equation and reads V φ = K + I u D. 3.4 Moreover, we consider Symm s integral equation in a more general way, i.e. we solve V φ = f 3.5 for an aritrary function f H Γ. Defining the ilinear-form a on H Γ H Γ y aφ, w := V φ, w leads to the equivalent variational formulation: aφ, w = f, w w H Γ. 3.6 Since the single layer operator V is H -elliptic, the lemma of Lax-Milgram implies the existence of a unique solution. With the triangulation T h of Γ and the polynomial degree p := p N N el we otain the discrete prolem: Find φ p h S T h, p, such that w p h S T h, p there holds aφ p h, wp h = f, wp h. 3.7 Note that we can apply the Céa lemma since the single layer operator is H - elliptic and thus, 3.7 has a unique solution. Choosing a asis Φ j j=,...,dim S T h,p we can rewrite 3.7 to a system of linear equations. Therefore, we define the coefficient vector x of φ p h with respect to asis Φ i and the right-hand side := i i=,...,dim S T h,p with i = f, Φ i. Then, 3.7 reads Vx =, where V denotes the Galerkin matrix of the single layer operator. In the special case of the Dirichlet prolem 3.4, we define the mass matrix M := M i,j j=,...,dim S T h,p i=,...,dim S T h,p with M i,j := Ψ j, Φ i and assume u D to e the coefficient

26 Chapter 3: The hp-bem for the Navier-Lamé Equation vector of u D,h with respect to the asis Ψ j. Using the Galerkin matrix of the doule layer operator we otain the following system: Vx = K + M u D. Concerning the a priori error analysis we state a result that is proven in [5], page 6: Theorem 3.. Dirichlet prolem Let T h e a triangulation of Γ with h := max{ Γ l : Γ l T h }. Moreover, let s and φ H s Γ e the exact solution of the Dirichlet prolem 3.4 and φ p h S T h, p the solution of the discrete prolem with polynomial degree p N on the discrete mesh T h. Then, there holds φ φ p h H h +min{s,p+} φ Γ H s Γ The Neumann Prolem and the Hypersingular Integral Equation In this section we solve the Neumann prolem { divσu = in Ω γ u = g on Γ. 3.9 We compute the missing Dirichlet data u H Γ with the second equation of the Caldéron system, namely the hypersingular integral equation, which reads W u = I K g. 3.3 Here, g H Γ denotes the Neumann data. variational formulation: To solve 3.3 we consider the Find u H Γ : W u, v = I K g, v v H Γ. 3.3 Since the hypersingular operator W is not H -elliptic, we cannot apply Lemma 3..6 Lax-Milgram and thus, 3.3 does not have a unique solution. In order to otain a prolem that has a unique solution we introduce some constraints in the following. We first consider the homogeneous integral equation, i.e. g in 3.3. Let R := span{v, v, v 3 } 3.3 with v =, T, v =, T and v 3 = x, x T. Plugging in v k k =,..., 3 into the second equation of the Caldéron-system 3.6 yields W v k =, where

27 Section 3.3: The Neumann Prolem and the Hypersingular Integral Equation we use the fact that γ v k = and σv k =. Thus, R is the solution space of the homogeneous integral equation. Note that in linear elasticity, v and v correspond to the translations of the ody in x - and x -direction, respectively, and v 3 corresponds to the rotation of the ody and consequentially, R is the set of all rigid ody motions in two dimensions. Thus, the solution of the non-homogeneous integral equation is unique except for the rigid ody motions. { In order to otain a prolem with a} unique solution, we define the space H R Γ := v H Γ : v, w =, w R of all H Γ -functions that are orthogonal to the rigid ody motions and consider the variational formulation Find u H R Γ : W u, v = K g, v v H R Γ Lemma 3.3. i The hypersingular operator W is H R Γ -elliptic, i.e. c > : W v, v c v v H R Γ. ii Let g H Γ e the given Neumann data. Then, for all v R there holds γ v, g =, 3.34 which is a constraint to the Neumann data for the solvaility of the prolem. Proof. adi See [6], page 58. adii The second Betti formula which is proven in [6], page 8, reads divσu T v dx + γ v T γ u ds x Ω Γ = divσv T u dx + γ u T γ v ds x. Ω Γ 3.35 Let u e the solution of 3.9 with divσu = and γ u = g in a weak sense and v R. Plugging in u and v into the second Betti formula yields γ v, g =. Due to i in Lemma 3.3. we are in the framework of the Lax-Milgram lemma and thus, 3.33 has a unique solution u H R Γ. However, instead of discretizing H R Γ which is numerically difficult due to the restrictions w, v =, w H Γ

28 Chapter 3: The hp-bem for the Navier-Lamé Equation and v R, we use the restrictions to uild a stailization of the hypersingular operator. We define for u, v H Γ u, v W +S := W u, v + 3 u, v k v k, v, where v k denote the translations in x - and x -direction and the rotation, respectively, and solve the modified variational prolem Find u H Γ : u, v W +S = K g, v v H Γ In the following lemma we proof the equivalence of 3.33 and Lemma 3.3. The variational formulations 3.33 and 3.36 are equivalent. Proof. First, we proof that if u is the solution of 3.36 we get u H R Γ. Since {v k, k =,..., 3} is a non-orthogonal asis of R we choose {w, w, w 3 } R to e an orthonormal asis of R. There holds 3 v k = a j,k w j k =,..., 3, 3.37 j= where the matrix A = a j,k j,k=,..,3 is regular. Moreover, there holds I K g, w j = g, w j }{{} I + K g, w j = = g, I + K w j =, where we used ii in Lemma 3.3. to otain the second identity, and the first equation of the Caldéron System and γ w j = to otain the last identity. Applying k= the second equation of the Caldéron System we get Testing 3.36 with w j we otain u, w j W +S = W u, w j + which reduces to W u, w j =. 3 u, v k v k, w j = K g, w j k= 3 u, v k v k, w j =, j =,.., 3 k= u, v A u, v =. u, v 3

29 Section 3.3: The Neumann Prolem and the Hypersingular Integral Equation 3 Since A is regular we have u, v k =, k =,..., 3, which implies u H R Γ. Additionally, if u H R Γ solves 3.33, 3.36 reduces to W u, v = K g, v. Thus, oth prolems are equivalent. The discrete prolem of 3.36 reads with p = p Find u p h S T h, p : u p h, vp h W +S = K g h, v p h vp h S T h, p, where g h denotes the projection of the Neumann data to S T h, p. In order to solve the discrete prolem we define the stailization matrix S := S i,j i,j=,..,p+ with S i,j := 3 Ψ j, v k v k, Ψ i. k= Moreover, let M e the mass matrix and K e the Galerkin matrix of the doule layer operator. Defining the coefficient vectors g of g h with respect to the asis of S T h, p and u of u p h with respect to the asis of S T h, p the system of linear equations reads W + Su = M KT g. Besides the Neumann prolem we also consider the hypersingular equation in a more general way, i.e. we solve W u = h 3.38 for an aritrary function h H Γ. For solving 3.38 we proceed as aove, i.e. the corresponding system of linear equations with stailization reads W + Su =, where := i i=,...,dim S T h,p with i := h, Ψ i. Concerning the a priori error analysis we state a result that is proven in [5], page 7: Theorem Neumann prolem Let T h e a triangulation of Γ with h := max{ Γ l : Γ l T h }. Moreover, let s and ψ Hs pwγ e the exact solution of the Neumann prolem 3.3 and ψ p h S T h, p the solution of the discrete prolem with polynomial degree p N on the discrete mesh T h. Then, there holds ψ ψ h H Γ h min{s,p+} ψ H s pw Γ. 3.39

30

31 Chapter 4 Associated Legendre Functions and Related Functions In this chapter, we estalish a asis for the assemly of the Galerkin matrices. As it is shown in Chapter 5, we can reduce the calculation of the Galerkin entries to the calculation of doule integrals of a specific type. Therefore, we investigate the calculation of these integrals, here. In the first section, we give the theory we need for the calculation of the doule integrals. We introduce the Legendre polynomials, the Loatto shape functions and state some important properties. Note that we use the Legendre polynomials and the Loatto shape functions for the definition of the asis functions of the ansatz- and test-spaces S T h, p and S T h, p in order to derive the hp-bem for the Navier- Lamé equation for an aritrary polynomial degree p. Furthermore, we introduce the associated Legendre functions and related functions that we need to discuss the calculation of the doule integrals in the second section. 4. Associated Legendre Functions and Loatto Shape Functions First, we define the Legendre polynomials y a three-term recurrence relation. Definition 4.. The Legendre polynomials are defined for z C y P k+ z = k + k + z P kz k k + P kz, k. 4. with the initial values P z =, P z = z. Note that this definition shows one of the most important advantages of the Legendre polynomials. Due to the three-term recurrence relation it is possile to evaluate the Legendre polynomials P k z with complexity Ok. Moreover, the computation is well-conditioned for z <, especially for z [, ] see []. Some elementary properties are stated in the following lemma. Lemma 4.. i The Legendre polynomials are orthogonal with respect to the L scalar product, P k t P m t dt = k + δ km, k, m N. 4.

32 6 Chapter 4: Associated Legendre Functions and Related Functions ii For all k N there holds P k = and P k = k. iii The antiderivative of the Legendre polynomial P k is given y iv There holds and t P k ξ dξ = k + P k+t P k t, k. 4.3 P k s P m t s t P k s P m t ds dt = 4δ k δ m, k, m N 4.4 4, m k odd ds dt = k + m + m k else 4.5 Proof. A proof of i, iii and iv can e found in [], and ii is proven in [3] page 79. Property iii in Lemma 4.. motivates the definition of the Loatto shape functions as antiderivatives of the Legendre polynomials. Definition 4..3 Let k 3 and t [, ]. Then, the Loatto shape functions are defined y N t = t, N t = + t t, N kt = P k ξ dξ. Note that in the literature, e.g. in [7], the Loatto shape functions N k are scaled with the factor P k. However, for our application to integral equations we omit this factor. The following properties can e derived immediately from the properties of the Legendre polynomials. Lemma 4..4 i The Loatto shape functions are related to the Legendre polynomials y N k t = k 3 P kt P k 3 t, k 3 ii The Loatto shape functions fulfill the three-term recurrence relation N k+ t = k 3 k t N k t k 3 k N k t, k 4 with the initial values N 3 t = t, N 4 t = t t

33 Section 4.: Associated Legendre Functions and Loatto Shape Functions 7 iii There holds N k ± =, k 3. iv For m 3, there holds P k s N m t 8, for m k even m km+km k 3m+k ds dt = s t, otherwise, for m = we otain and for m = we get P k s N t s t P k s N t s t ds dt = ds dt = kk+ kk+ kk+ kk+, for k odd, otherwise, for k odd, otherwise. Proof. The definition of the Loatto shape functions and Lemma 4.. iii imply i. For the proof of the three-term recurrence relation ii see []. iii results from Lemma 4.. ii and from i. The representation of the Loatto function of i and Lemma 4.. iv imply iv. For the introduction of the Legendre functions we are guided y the works of [] and [3] and summarize the results that are relevant for the application to the hp- BEM. Let us consider the associated Legendre differential equation which is given y d dz { z du dz } + [kk + m z ] u =, 4.6 where we assume m, k N and z C\[, ]. We start investigating 4.6 for m =. In this case, 4.6 is called the Legendre differential equation and all solutions of 4.6 are called Legendre functions. The Legendre polynomials that we introduced aove solve the Legendre equation and hence, they are also called the Legendre functions of first kind. It is shown in [] page 5 that every solution of 4.6 has the following representation uz = A P k z + B P kz log z + z W kz, A, B C 4.7 with polynomials W k z that are defined y W z =, W z =, W k z = k + k + z W kz k k + W k z. 4.8 This motivates the definition of the Legendre functions of second kind Q k y Q k z := P kz log z + z W kz. 4.9 An important relationship to the Legendre polynomials is given y Neumann s formula which is stated in the following theorem.

34 8 Chapter 4: Associated Legendre Functions and Related Functions Theorem 4..5 Let k N and z C\[, ]. Then, there holds Q k z = P k t dt. 4. z t Proof. See [], page 63/67. Considering the general case m N we define the associated Legendre functions as solutions of the associated Legendre equation. The associated Legendre functions of second kind Q m k z are given y Q m k z := z m d m dz m Q kz. 4. and solve 4.6. With this definition we can generalize Neumann s formula as follows. Theorem 4..6 Let k, m N and z C\[, ]. Then, there holds Q m k z := m m! z m P k t dt. 4. z t m+ Proof. See [], page 6. Since we only need the integral in 4. for the application to the hp-bem, we give the following definition. Definition 4..7 Let k, m N and z C\[, ]. We define Q m k z := Q k z := Q k z := P k t dt 4.3 z t m+ P k t logz t dt 4.4 P k t z t logz t dt. 4.5 Referring to [] and [] we summarize the properties of the function Q m k z without a proof. Lemma 4..8 Let z C\[, ]. i For m and k max{, m}, there holds the three-term recurrence relation k m + Q m k+z = k + z Q m k z k + m Q m kz.

35 Section 4.: Special Integrals Related to the Associated Legendre Functions 9 ii For k N there holds the following symmetry property Re Im Re Im Q k z = k Re Q k z = k Im Q z = k Re Q k z Q k z Q Q z = Im Q z z +π i, for < argz < π π i, otherwise. 4.6 iii The functions Q k the limits lim z Q k z and Q z can e continuously extended to z =, i.e. k z and lim z Q k z exist. iv The function Q z is not one-valued for all z, ], i.e the imaginary part of Q z has a jump. v Q k can e represented y Q kz = P k z log z + z W kz, 4.7 where the polynomials W k are defined y Special Integrals Related to the Associated Legendre Functions In this section we first define some special integrals we need for the calculation of the Galerkin matrices. Referring to [9], we introduce the following notations. Definition 4.. Let a, C, with a + [, ] and a [, ]. Then, we define for j, k, p N I j,k a, := I p j,k a, := and for j, p N and k N O j,k a, := O p j,k a, := P j s P k t loga + s t dt ds, 4.8 P j s P k t dt ds 4.9 a + s t p+ P j s N k t loga + s t dt ds, 4. P j s N k t dt ds. 4. a + s t p+ Moreover, we extend this definition and introduce two more integrals we need for the calculation of the Galerkin matrices.

36 3 Chapter 4: Associated Legendre Functions and Related Functions Definition 4.. Let a, C, with a + [, ] and a [, ]. Then, we define for j, k, p N Ĩ j,k a, := Ĩ p j,k a, := and for j, p N and k N Õ j,k a, := Õ p j,k a, := P j s P k t Ima + s loga + s t dt ds, 4. P j s P k t Ima + s dt ds, 4.3 a + s t p+ P j s N k t Ima + s loga + s t dt ds, 4.4 P j s N k t Ima + s dt ds. 4.5 a + s t p+ Since we only need I j,k, Ĩ j,k, O j,k and Õ j,k for the application to the BEM we restrict to investigate these integrals. The calculation via recurrence relations and the stale and efficient implementation of I p j,k and Op j,k p N is descried in [9] for a ± [, ]. However, for calculating the Galerkin matrices we get the special case a ± = ± considering neighoring elements see Section 5.. Therefore, we continuously extend the integrals to a ± = ± and otain the following lemma. Lemma 4..3 Let a n n N, n n N C\[, ] with lim n a n = a C, lim n n = C and a ± = ±. Then, there holds for j N, k N i lim n I j,k a n, n = I a, j,k ii lim n Ij,k a n, n = Ij,k a, iii lim n Ĩj,k a n, n = Ĩ j,k a, iv lim n Oj,k a n, n = Oj,k a, In particular, all limits in i iv exist. Proof. Since all integrals are calculated via recurrence relations we only investigate the initial values. adi See []. adii According to [9] the initial values of Ij,k a n, n are calculated with Q k a n + n, Q k a a n n and Q n a k n, Q n k n, respectively. Note that a+ = ± ± a = and a = ± ± a =. Therefore, we have to investigate the limit lim Q z ± z in order to proof ii. Due to Lemma k

37 Section 4.: Special Integrals Related to the Associated Legendre Functions iii the limit exists for z. For z, we use the symmetry of Q k z Lemma 4..8 ii which yields lim z Q k z = k+ lim Q z k where κ = ± depends on the argument of z. adiii With the linearity of the integral we get Ĩ j,ka n, n = Ima n I j,ka n, n + Im n z + κ π i, s P k t P j s ds dt a n + n s t Using the three-term recurrence relation of the Legendre polynomials 4. yields for j s P j s = j + j + P j+s + j j + P js and s P s = P s, respectively. Thus, there holds for j = and Hence, [ j + j + s P k t P j s ds dt a n + n s t P k t P j+ s ds dt + j a n + n s t j + s P k t P s ds dt = a n + n s t ] P k t P j s ds dt a n + n s t P k t P s ds dt. a n + n s t Ĩj,ka n, n = Ima n Ij,ka n, n [ j + + Im n j + I j+,ka n, n + j ] j + I j,ka n, n and 4.6 With ii we otain iii. Ĩ,ka n, n = Ima n I,ka n, n + Im I,ka n, n. 4.7 adiv With the definition of the Loatto shape functions N and N and Lemma 4..4 i we get for k 3 Oj,a n, n = I j, a n, n Ij,a n, n Oj,a n, n = I j, a n, n + Ij,a n, n Oj,ka n, n = I k 3 j,k a n, n Ij,k 3a n, n. Applying ii completes the proof.

38 3 Chapter 4: Associated Legendre Functions and Related Functions With this result we see that we can use the results of [9], 4.6 and 4.7 for the calculation of I j,k, Ĩ j,k and O j,k. Therefore, we do not go into detail aout the calculation of these integrals. It still remains to investigate the integral Õj,k for the calculation of the Galerkin matrices. According to [9], the initial values are calculated y Q k z, which cannot e extended to z =. Thus, we cannot prove the existence of an extension to a ± = ± in the same way as in Lemma In the following, we first discuss the calculation of Õ j,k for a ± [, ], where we derive formulas that are different from the formulas in [9], and then we show that Õj,k can e extended to a ± = ±. By analogy with 4.6 and 4.7 we get [ j + Õj,ka, = Ima Oj,ka, + Im j + O j+,ka, + j ] j + O j,ka, 4.8 and Õ,ka, = Ima O,ka, + Im O,ka,. 4.9 Hence, we discuss the calculation of Oj,k a, which is done in the following way:... O,k... O,k Oj, Oj, Oj,3... Oj,k This structure for the computation results from the fact that the Loatto shape functions N and N do not fulfill a recurrence relation, and thus we have to compute the first an the second column, separately. Moreover, we calculate the Oj,k a,, j, k 4, via the four-term recurrence relation which is given in [9] Theorem 9 y O j,ka, = a j j + k O j,ka, j k j + k O j,ka, + j j + k O j,ka,. The initial values for this recurrence relation are O,k, O,k and O j,3. Thus, these values have to e calculated first. We start investigating the first and second column and otain the following lemma.

39 Section 4.: Special Integrals Related to the Associated Legendre Functions 33 Lemma 4..4 Let a, C, with a ± [, ] and j N. Then, there holds O,a, = { } Q a a η Q η + Q a { Oj,a, a = Q j+ j + Q a j } Q a j+ + Q a j + Q a j and with O,a, = O j,a, = { Q j + } a a η Q η Q a { a Q j+ Q a j } Q a j+ + Q a j Q a j, for conv { a η =, } a {x R : x } =, otherwise. 4.3 Proof. We start proving the representation of O, and O,. a ± [, ], there holds For a, C with = = + t ds dt a + s t log a s + t + a log a + s a + s For the first integral we otain with loga = loga + log + a + s a + s t a s log a + s a + s + ds. ds t= s ds { π i, for arga + arg π + π i, for arga + arg < π depending on the arguments of the complex numers a, a a log = = Q a log a a s log s Q a log a s s ds + + κ 4πi, and ds κ πi ds

40 34 Chapter 4: Associated Legendre Functions and Related Functions Q where κ {,, }. Note that instead of adding κ 4πi explicitly we use the symmetry property of in Lemma 4..8 ii which yields a a log s log a a = Q η Q η, with s ds, for conv { a η =, } a {x R : x } =, otherwise. For the second integral we otain With a + s a + s a + s a + s + ds = = s = z s Thus, we have With = Q a + s a s a + s a a s ds s a s a s + a z, z C, we simplify 4.3 and get z s t ds dt a + s t a η a + s a + s a + s a + s + ds = a s a s ds = a Q Q a. Q = = = a η + Q dt ds a + s t a + s a + s + ds Q a s a s ds a Q a s s a ds. 4.3 s a Q a.

41 Section 4.: Special Integrals Related to the Associated Legendre Functions 35 and the definition of N t = t and N t = + t we get the formula for O,a, and O,a,. For j N we use the following result which is proven in [9] Theorem 6: I j,a, = j j + I j+,a, + j + j + I j,a, + a I j,a, With the definition of N t and 4.33 we otain Oj,a, = I j, a, Ij,a, = a I j j,a, j + I j+,a, j + j + I j,a,. With the initial values of Ij,k [9], Lemma comined with Lemma 7: Ij,a, = a Q Q a we otain Oj,a, = { [ ] a a Q j Q a j j [ ] a Q j+ j + Q a j+ j + [ ]} a Q j+ j + Q a j Applying the three-term recurrence relation of Q k Lemma 4..8 i, which reads z Q jz = j + j + Q j+z + j j + Q jz, to Q a j and a Q j, we otain { Oj,a, a = Q j+ j + Q a j Q a j+ } + Q a j + Q a j. Note that we added and sutracted Q a j. The proof of the formula for Oj,a, can e done similarly and is omitted here. The formula for the calculation of the third column is given in the next lemma. Lemma 4..5 Let a, C, with a ± [, ], j and η as defined in 4.3. Then, there holds [ ] [ ] O,3a, = a Q a a η Q η + Q a a η Q η a [ ] a 3 Q Q a [ ] a a Q Q a 3 [ ] a Q Q a +,

42 36 Chapter 4: Associated Legendre Functions and Related Functions [ O,3a, = a Q a η a η Q ] a η + 3 [ ] a η + [ 3 Q Q a 3 [ ] a 5 Q Q a [ ] a 3 a Q Q a 3 [ a a Q [ ] a 5 Q 3 Q a 3 +, Q a η Q ] a Q η ] a and { } Oj,3a, j a = a Q j+ j + j + Q a j+ [ a ] { } + j + + j a Q j j + j Q a j { } 3j + a + a Q j j + j + Q a j { } j a + Q j j + j Q a j. Proof. The proof can e found in Appendix A. The formulas for the calculation of the first and the second row of 4.3 is given in the following lemma. Lemma 4..6 Let a, C, with a ± [, ] and k 3. Then there holds O,k a, = [ Q k 3 k a Q k a + Q k 3 a + Q ] k 3 a + and { O,k a, = Q k 3 k η a + Q Q k η a Q k 3 η a + + k 3 η a [ Q k a + + Q k a Q k 3 a + Q ] } k 3 a. with, for conv {a +, a } {x R : x } = η =, otherwise. 4.35

43 Section 4.: Special Integrals Related to the Associated Legendre Functions 37 Proof. We start with proving the representation of O,k. There holds O,k = N k t ds dt a + s t = N k t a + t s ds dt = N k t a + t a + t dt = N k t a + t dt. a t Applying the representation of the Loatto shape functions in Lemma 4..4 i and using the definition of Q k we get the formula for O,k. For the second identity we otain analogously O,k = N k t s ds dt a + s t = s N k t a + t s ds dt = N k t log s + a t + t a a + t s dt s= = { N k t log + a t log + a t + t a } a + t a + t + dt As it is done in the proof of Lemma 4..4 we simplify the logarithm terms in 4.36 y { N k t log + a t log + a } t dt { } = N k t log a + t log a t dt = N k t {log a + t log a t} dt + κ πi N k t dt, where κ {,, }. Since N kt dt = P k 3 kt dt P k 3t dt δ 3 k,3 we otain { N k t log + a t log + a } t dt = k 3 { Q k η a + Q k η a Q k 3 η a + + = } Q k 3 η a,

44 38 Chapter 4: Associated Legendre Functions and Related Functions where we define η y, for conv {a +, a } {x R : x } = η =, otherwise. Again, we added κ 4 3 πi δ k,3 implicitly y point reflection of a + and a using the Q symmetry property of. For the remaining terms in 4.36 we get { N k t t a } a + t a + t + dt a t = N k t a + t a t dt a t = N k t a + t dt a t = k 3 Note that we used Q k a + Q k a Q k 3 a + + Q k 3 a t = z t together completes the proof.. z, z C, for the second identity. Putting the results z t With the last three lemmas, 4.8 and 4.9 we can calculate Õ j,k a, provided that a± ±. However, it still remains to investigate the existence of an extension of Õ j,k for a ± = ± which is done in the following theorem. Theorem 4..7 Let a n n N, n n N C\[, ] with lim n a n = a C, lim n n = C and a ± = ±. Then, there holds for j N, k N lim Õj,ka n, n = n Õ j,ka,. Proof. To proof the existence of an extension we use the representation 4.7 of Q k which reads Q kz = P k z log z + z W kz. In particular, we divide the representation of Q k z into a singular term with coefficient P k z and a polynomial W k z. Thus, we only consider the singular terms in the formulas for the calculation of Õ j,k and show that they add up to zero. Since the way of proceeding is the same in all cases and since there are many cases due to the different formulas for the calculation of Õ j,k we restrict to investigate Õ,. In this case we consider a + = a + = a = a = a = a = a = a =.

45 Section 4.: Special Integrals Related to the Associated Legendre Functions 39 Let a n n N, n n N e as provided in the theorem. Then, we get Õ,a n, n = Ima { } n Q a n a n η Q η n n + Im { n Q an 6 Q an n } n Q an + Q an + Im n an Q n n n + Ima n n Q n. an n In the case a + = we define z n := an n For the singular terms there holds n to simplify the notation. [ Imn lim n 6 Q z n + Q z n [ = Im 6 lim n ] P z n + P z n }{{} n = Im W W. 3 log z ] n + z n + Im W W 3 Note that the Legendre polynomials vanish due to Lemma 4.. ii. In the case a + = we define z n := an n [ Iman lim Q n z n + Im n Q n z n + Im n n 6 = lim n Ima = Ima [{ Iman P z n + Im n P z n n }{{ n } Ima+ = n W Im W Im n. For the singular terms there holds + Im n 6 Q z n Q ] z n P z n P z n }{{} n W Im W W 3 W Im W W. 3 } log z ] n + z n In the case a = we define z n := an n n and otain [ Imn lim n 6 Q z n + Q z n [ = Im 6 lim n ] P z n + P z n }{{} n = Im W W. 3 log z ] n + z n + Im W W 3

46 4 Chapter 4: Associated Legendre Functions and Related Functions For a = we define z n := an n [ Iman lim Q n z n + Im n Q n z n + Im n n 6 = lim n Ima = Ima [{ Iman P z n + Im n P z n n }{{ n } Ima = n W Im W Im n and get + Im n 6 Q z n Q ] z n P z n P z n }{{} n W Im W W 3 W Im W W. 3 Thus, we otain finite values for Õ, a n ± n ±. } log z ] n + z n

47 Chapter 5 Realization of the hp-bem for the Navier-Lamé Equation In this chapter, we descrie the realization the hp-bem for the Navier-Lamé equation. We egin y introducing a parametrization and some notations that we need for the calculations in this chapter. The main point for implementing the hp-bem for high polynomial degrees is the choice of the asis functions of the discrete spaces S T h, p and S T h, p, which is descried in the second section. The calculation of the Galerkin matrices is given in the third section. We close this chapter y introducing some a posteriori error estimators that can e used to create adaptive algorithms. 5. Geometrical Basics In this section we introduce a parametrization of the oundary elements and state some geometrical results that we need for the calculation of the Galerkin matrices. Let Γ l T h e a oundary element. Defining the center m l := A l+b l of the oundary element Γ l we introduce the following parametrization [, ] Γ l γ l : t B l A l t + m l. Note that we otain γ l t = B l A l and γ l t = Γ l, and which implies γ l x = B l A l B l A l T x m l, γ l x = B l A l B l A l. 5. As we see in the susequent sections all calculations for the Galerkin matrices are performed for a fixed comination of oundary elements. Therefore, we consider the oundary elements Γ l, Γ m T h with x Γ l and y Γ m. The configuration is given in Figure 5.. Using the parametrization and the vectors u := Bm Am and v := B l A l we get x = m l + s v and y = m m + t u with s, t [, ]. In the following we take a closer look at the term x y. With w := m m m l

48 4 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation y m l w x u v Fig. 5.: Configuration of the oundary elements Γ l and Γ m. there holds x y = m l m m + s v t u = s v w t u and hence x y = s v w t u = s v w t u T s v w t u = u T u t s v w T u t + s v w T s v w = u T uzs tzs t, 5. where zs is one of the zeros of the quadratic equation with zs = s v wt u + [s v w T u] u T u s v w T s v w. u T u Applying the Lagrangian identity we get which yields [s v w T u] u T u s v w T s v w = [u s v w] zs = s v wt u + i u s v w. u T u To simplify the notation we define a := ut w i u w u T u and := ut v+i u v u T u and get zs = a + s. 5.3 Note that in the special case of identical elements Γ m = Γ l we get x y = s t u and zs = s. Moreover, one can show that in the case of neighoring elements, i.e. A l = B m or A m = B l, we get a ± = ±.

49 Section 5.: Basis Functions of the Discrete Function Spaces Basis Functions of the Discrete Function Spaces In order to calculate the Galerkin matrices it remains to choose an appropriate asis of the discrete spaces S T h, p and S T h, p. Therey, we do not choose the monomials, as it is done in the standard approach, ut we choose the Legendre polynomials and the Loatto shape functions, as they fulfill recurrence relations and can e evaluated efficiently up to a high polynomial degree. Additionally, this choice of the asis functions leads to recurrence relations the Galerkin entries can e computed with very efficiently. In the following we descrie the asis functions more precisely. Let T h e a triangulation of the polygonal oundary Γ with N c corner points and N el := T h. Additionally, let p N N el and p N N el. As the functions of S T h, p are piecewise polynomial, gloally discontinuous functions, we choose the transformed Legendre polynomials as asis functions, i.e. the asis is given y l P j x and P l, j x where the transformed Legendre polynomials are defined y x := P j γ l x for l =,..., N el and j =,..., p,l. Note that { } { } l P supp j x = supp P l = Γ l j x and thus, we have a local asis, i.e every asis function only lives on one oundary element. For the implementation we sort the asis functions as follows: P l j. The asis functions are sorted y their non-zero entries, i.e all asis functions whose first entry is non-zero are listed first.. The asis functions are sorted y the oundary element they live on. 3. The asis functions are sorted y their polynomial degree. All functions in S T h, p are gloally continuous and hence, we cannot choose the Legendre polynomials as asis functions. However, the Loatto shape functions N j t vanish at the endpoints ± for all j 3 and thus, we choose the transformed Loatto shape functions as asis functions for j 3 Ñ l j x and Ñ l, j x with Ñ l j x := N j γ x. Again, there holds supp l {Ñ l j x } { } = supp Ñ l j x = Γ l. However, the first two Loatto shape functions N and N do not vanish at the endpoints. Hence, to ensure the gloal continuity we add hat functions to the asis,

50 44 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation that live on two neighoring elements. Thus, we have oth linear asis functions that live on two elements and asis functions of higher polynomial degree that live on one element. For the implementation we introduce the following oder of the asis functions:. The asis functions are sorted y their non-zero entries, i.e all asis functions whose first entry is non-zero are listed first.. All hat functions are listed efore all other asis functions with higher polynomial degree. 3. The asis functions are sorted y the oundary element they live on. 4. The asis functions are sorted y their polynomial degree. Figure 5. shows an example of the asis functions for three elements and p = p =,, 3. In particular, the figure shows the enumeration of the asis functions of S T h, p within one component Fig. 5.: Basis function for S T h, p left and S T h, p right for three elements and p = p =,, 3. Source: [3]. Finally, we define the numer of degrees of freedom with respect to the discrete space S T h, p y N el N := dim S T h, p = p,k + k= and the degrees of freedom with respect to S T h, p y N := dim S T h, p = Nel p,k + N c. Considering different types of oundary conditions we denote the degrees of freedom on the Dirichlet oundary y N i D := dim Si T h,d, p i, on the Neumann oundary y N i N := dim Si T h,n, p i and on the gliding oundary susequently defined y N i G := dim Si T h,g, p i i =,. k=

51 Section 5.3: Computation of the Galerkin Matrices Computation of the Galerkin Matrices In the section we descrie the calculation of the Galerkin matrices. In particular, we derive analytical formulas that depend on the doule integrals that we introduced in Section 4.. Hence, we can assemle the Galerkin matrices efficiently and staly for high polynomial degrees. In the first section we investigate the Galerkin matrix of the single layer operator and in the second section we descrie the calculation of the Galerkin matrix of the doule layer operator. Finally, we derive formulas for the entries of the Galerkin matrix of the hypersingular operator The Single Layer Operator With the definition of the asis functions for S T h, p the Galerkin matrix for a fixed comination of oundary elements Γ l and Γ m with polynomial degrees p l and p m reads: V V V V P m P m.,, P m,. P m., l P V l P p l V m l m P p m P, P V, P l V P m p m l P V P p m, m l P p l V, P l p l V P m, P l p l V m P p m, P l P m p m l P V P m, P m p m, P l p l V P m,, P p l l P l V P p m, m P l P l p l V P m p m, P p l l Using the given lock structure of the Galerkin matrix, we compute the matrix of the j, k-th entry j =,..., p l and k =,..., p m of each lock, since there holds m l m P k P j P k V, V, P l j l P V P m j = l m P j x Ux, y P k y ds y ds x. Γ, V k P m, k P l l Γ } m {{} j =: V l,m j,k For the calculations, we distinguish the case of identical elements Γ l = Γ m and non-identical elements Γ l Γ m = and investigate oth cases, separately. Starting the calculation of V l,m j,k for identical elements, we need the following result that is proven in []. Lemma 5.3. Let Γ l T h e a oundary element with h l := Γ l and k, j N. Then, there holds Γ l P l j x log x y Γ l l P k y ds y ds x h l log h l 3, for j = k = h = l, for j + k > and j k even j+k+j+kj k, otherwise..

52 46 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation With Lemma 5.3. we otain the following explicit formula in the case of identical elements. Theorem 5.3. Let Γ l T h e a oundary element and u e the directional vector of Γ l as defined in Section 5.. Moreover, let j, k N. Then, there holds in the case of identical elements V l,l j,k = λ + 3µ 4πµλ + µ ut u θ j,k I λ + µ λ + 3µ 4δ uu T k δ j, u T u where log 4 v T v 6, for j = k = θ k,j = 8, for j + k > and j k even j+k+j+kj k, otherwise. Proof. For reasons of clarity we split the Kelvin matrix 3. into the two parts U and U with Ux, y = λ + 3µ 4πµ λ + µ log x y I }{{} =: U x, y λ + µ λ + 3µ x yx y T } x y {{ } =: U x, y 5.4 and investigate oth parts, separately. We get l l P j x Ux, y P k y ds y ds x Γ l Γ l = λ + 3µ { l P j x U x, y 4πµ λ + µ Γ l Γ l λ + µ λ + 3µ Γ l l P j x l P k y ds y ds x Γ l U x, y Using Lemma 5.3. there holds for the first integral l P j x U l x, y P k y ds y ds x Γ l Γ l l l = P j x log x y P k y ds y ds x I = θ k,j I Γ l Γ l P l k y ds y ds x } with log 4 u T u 6, for j = k = θ k,j := 8, for j + k > and j k even j+k+j+kj k, otherwise.

53 Section 5.3: Computation of the Galerkin Matrices 47 Note that we used the fact that h l = u. For calculating the second term we plug in the parametrization and use x y = s t u, which we already stated in Section 5.. Thus, we otain Γ l = l P j x U x, y Γ l Γ l = Γ l 4 l P k y ds y ds x x yx y Γl T l l P x y k y P j x ds y ds x = 4δ k δ j uu T. P k tp j sds dt uut u T u Note that we used iv in Lemma 4.. and u T u = u = Γ l Putting the results together completes the proof. 4 for the last identity. In the case of non-identical elements we reduce the calculation of V l,m j,k to the calculation of the integrals I j,k and Ĩ j,k that we introduced in Chapter 4. For reasons of clarity we split the Kelvin matrix into two parts and investigate oth parts separately, as it is already done in the proof of Theorem 5.3. See 5.4. We start investigating the first part of the Kelvin matrix and otain the following lemma. Lemma Let Γ l, Γ m T h e oundary elements with Γ l Γ m =. Moreover, let u and v e the directional vectors of Γ l and Γ m and zs = a + s as defined in Section 5.. Then, there holds for j, k N l P j x U x, y Γ m Γ l m P k y ds y ds x = u v { logu T u δ k δ j + Re I j,k a, } I. Proof. We start plugging in the parametrization and otain with u = Γ l Γm, v = and z := zs = a + s Γ l = = = = P l j Γ l u v u v u v x U x, y Γ m log x y Γ m logu T u m P k y m P k y ds y ds x l P j xds y ds x I log s v w t u P k t P j s dt ds I logu T uz tz t P k t P j s dt ds I P k t P j s dt ds + logz t P k t P j s dt ds + logz t P k t P j s dt ds I. 5.5

54 48 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation Since there holds log z = log z, z C, we otain and hence, logz t P k tp j s dt ds = logz t P k tp j s dt ds logz t P k tp j s dt ds + logz t P k tp j s dt ds = Re logz t P k tp j s dt ds. With the definition of I j,k Γ l l P j x and iv in Lemma 4.. we can write 5.5 shorter Γ m U x, y m P k y ds y ds x = u v { logu T u δ k δ j + Re I j,k a, } I. Before we investigate the second part of the Kelvin matrix we state the following partial fraction decompositions that we need for the calculations. Lemma Let t [, ] and z C\[, ]. Then, there holds i ii iii t z tz t = Rez Imz i Imz t z tz t = z tz t = i Imz z t + z t z t z t z t z t Rez i Imz. Rez z t z t z t + z t Proof. The proof can e done y multiplying the equations with z tz t and comparing the coefficients. For the second part of the Kelvin matrix we otain the following result. Lemma Let Γ l, Γ m T h e oundary elements with Γ l Γ m =. Moreover, let u and v e the directional vectors of Γ l and Γ m and zs = a + s as defined in Section 5.. Then, there holds for j, k N l P j x U m x, y P k y ds y ds x Γ l Γ m uu = u v {4δ T uu k δ j Ĩ T T u T u + Im u u j,k a, u T u } u u Re Ĩ T T + uu j,ka,. u T u 5.6

55 Section 5.3: Computation of the Galerkin Matrices 49 Proof. First, we investigate U x, y. With the parametrization and z := zs we otain U x, y = = = x yx yt x y sv w + t usv w + t ut u T uz tz t t u T uz tz t uut + t u T uz tz t u T uz tz t s v ws v wt. s v wu T + us v w T In general, every vector R can e represented y a linear comination of u and u := u, u T, i.e. Applying this identity on s v w yields = ut u T u u u u T u u. u T s v w s v w = u u T u = Rez u Imz u, u s v w u T u u where we used the representation of zs in 5.3. Hence, we get t U x, y = u T uz tz t uut t u Rez u Imz u u T T + Rez u Imz u u T uz tz t + u T uz tz t Rez u Imz u Rez u Imz u T, which we can further simplify y applying Lemma as follows U x, y = [ { u T Rez Imz u i Imz z t z t { + z t + + Rez z t i Imz z t z t + Rez u Imz u u T i Imz = uut u T u + Imz i Imz Rez z t + } uu T z t } u Rez u Imz u T z t ] Rez T u Imz u Rez u Imz u z t z t uu T u u T z t u T u z t + z t u u T + uu T u T. 5.7 u

56 5 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation With this representation of U and the linearity of the integral we otain l P j x U m x, y P k y ds y ds x Γ l Γ { m = u v P j s P k t dt ds uut u T u + P j s P k t Imz i z t dt ds uut T u u z t u T u P j s P k t Imz z t + dt ds u } u T T + uu. z t u T u In general, for all z C there holds z = z which implies that P j s P k t Imz z t dt ds z t P j s P k t Imz = dt ds z t P j s P k t Imz = i Im dt ds z t P j s P k t Imz z t dt ds 5.8 and P j s P k t Imz z t + dt ds z t P j s P k t Imz P j s P k t Imz = dt ds + dt ds z t z t P j s P k t Imz = Re dt ds. 5.9 z t With the definition of Ĩ j,k a, and Lemma 4.. iv we finally have l P j x U x, y Γ l Γ m uu = u v {4δ T k δ j Re Ĩ j,ka, m P k y ds y ds x uu Ĩ T T u T u + Im u u j,k a, u T u } u u T T + uu. u T u Putting the results of the previous lemmas together we get a formula for the computation of the single layer operator. Theorem Let Γ l, Γ m T h e a oundary elements with Γ l Γ m = and u and v e the directional vectors of Γ l and Γ m and zs = a+ s as defined in Section

57 Section 5.3: Computation of the Galerkin Matrices Moreover, let j, k N. Then, there holds V l,m j,k = λ + 3µ { [ ] u v logu T u δ kδ j + Re I 4πµλ + µ j,k a, I λ + µ uu [4δ T uu k δ j Ĩ T λ + 3µ u T u + Im u u T j,k a, u T u u u Re Ĩ T ]} + uu T j,k a, u T. u The Doule Layer Operator The Galerkin matrices of the single and the doule layer operator have the same lock structure, as the asis functions of S T h, p and S T h, p are sorted similarly y their components. Therefore, we compute the entries of the Galerkin matrix of the doule layer operator similarly to the entries of the matrix of the single layer operator. Hence, we calculate for a fixed comination of oundary elements Γ l and Γ m with polynomial degrees p,l and p,m j =,..., p,l and k =,..., p,m : K K Ñ m k Ñ m k,, l P j P l j K K Ñ m k Ñ m k, P l j =, P l j Γ l l P j x Γ m T x, y Ñ m k y ds y ds x. } {{ } =: K l,m j,k Again, we distinguish the cases of identical elements and non-identical elements. In the case of identical elements we derive the following explicit formula. Theorem Let Γ l T h e a oundary element and u e the directional vector of Γ l as defined in Section 5.. Moreover, let j N and k N. Then, there holds in the case of identical elements K l,l j,k = µ πλ + µ u η j,k I I where for k 3 8, for j k even k jk+jk j 3k+j η j,k :=, otherwise, and η j, :=, for j odd, otherwise, η j, := jj+ jj+ jj+ jj+, for j odd, otherwise. Proof. In the case of identical elements there holds x y T ny = s t u T ny =. Plugging in the parametrization we otain for the co-normal derivative T x, y of the Kelvin matrix that is given in 3.3 with t := ty = u u T x, y = = µ x y T t πλ + µ x y µ πλ + µ u s t I I.

58 5 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation and thus K l,l j,k = Γ l = l P j x T x, y Ñ l k Γ l µ πλ + µ u y ds y ds x N k t P j s s t ds dt I I. Simplifying the notation we define η j,k := N k t P j s ds dt. Thus, we otain s t K l,l j,k = With iv in Lemma 4..4 we get for k 3 µ πλ + µ u η j,k I I. 8, for j k even k jk+jk j 3k+j η j,k =, otherwise, and η j, =, for j odd, otherwise, η j, = jj+ jj+ jj+ jj+, for j odd, otherwise, which completes the proof. In the case of non-identical elements, we derive a formula for the computation of the Galerkin entries that depends on the integrals Oj,k a, and Õ j,k a,. To simplify the calculations, we split T x, y into three parts: µ x y T ny µ x y T ty T x, y = I + I I πλ + µ x y }{{ πλ + µ x y }}{{ } =: T x, y =: T x, y + λ + µ x y T ny x yx y T. πλ + µ x y } 4 {{} =: T 3 x, y 5. In the following we successively investigate all parts stating the next three lemmas. In the of this section, we summarize the results in Theorem Lemma Let Γ l, Γ m T h e oundary elements with Γ l Γ m =. Moreover, let u and v e the directional vectors of Γ l and Γ m and zs = a + s as defined in Section 5.. Then, there holds for j N and k N Γ l l P j x T x, y Ñ m k y ds y ds x = v Im Oj,ka, I 5. Γ m

59 Section 5.3: Computation of the Galerkin Matrices 53 Proof. First, we investigate T x, y. Plugging in the parametrization we otain with n := ny, z := zs and 5. where we used T x, y = x yt n x y I s v w T n u T u = s v w t ut n u T u z tz t I = s v wt n u T u = Imz u = z tz t I z tz t I, s v w u u T u u = Imz. u With the partial fraction decomposition in Lemma iii we get T x, y = i u z t. 5.3 z t Thus, we otain for the integral l P j x T x, y Ñ m k y ds y ds x Γ l Γ m = u v N k t P j s i u z t ds dt I z t N k t P j s = v Im ds dt I. z t Note that we otain the last identity with 5.8 y sustituting P k t with N k t. Plugging in the definition of Oj,k a, yields 5.. Lemma Let Γ l, Γ m T h e oundary elements with Γ l Γ m =. Moreover, let u and v e the directional vectors of Γ l and Γ m and zs = a + s as defined in Section 5.. Then, there holds for j N and k N l P j x T x, y Ñ m k y ds y ds x = v Re Oj,ka, I I. 5.4 Γ m Γ l Proof. As in the proof of the previous lemma we start with investigating T x, y. With the parametrization, ty = u, z := zs and 5. there holds u T x, y = x yt ty x y I I = s v w t ut u u u T u z tz t I I = { s v w T u u u T u z tz t t z tz t } I I

60 54 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation Applying i and ii in Lemma we get T x, y = { [ Rez u i Imz z t + z t z t + ] z t Rez i Imz z t } I I z t = u z t + I I. 5.5 z t Hence, y using 5.9 we otain l P j x T x, y T Ñ m k y ds y ds x Γ l Γ m = u v N k t P j s u z t ds dt I I z t N k t P j s = v Re ds dt I I. z t Plugging in the definition of Oj,k a, completes the proof. Lemma 5.3. Let Γ l, Γ m T h e oundary elements with Γ l Γ m =. Moreover, let u and v e the directional vectors of Γ l and Γ m and zs = a + s as defined in Section 5.. Then, there holds for j N and k N Γ l = v l P j x T 3 x, y T Ñ m k y ds y ds x Γ m { Im Oj,k a, I Re Õ j,k a, u u Im Õ T + uu T } j,k a, u T u Proof. We start simplifying T 3 x, y. Since there holds T 3 x, y = T x, y U x, y uu T u u T we use the representations 5.7 and 5.3 and otain T 3 x, y = i u z t { uu T z t u T u + Imz i z t uu T u u T z t u T u Imz u u T + uu T } = { u i Imz 4i u T u 5.6 z t + z t u T u z t uu T z t u T u Imz 4 z t uu T u u T z t u T u z t z t z t + u u T + uu T } z t u T. u With z t = z t z t + + z t i Imz z t z t

61 Section 5.3: Computation of the Galerkin Matrices 55 and z t z t z t + = z t z t z t we get T 3 x, y = { u i z t uu T z t u T u Imz 4 4i z t z t u T u Imz = { u 4i uu T u u T 4i z t u u T + uu T } z t u T u z t uu T + u u T z t u T u Imz 4 z t + uu T u u T z t u T u Imz 4i z t u u T + uu T } z t u T. u Simple calculus shows that uut +u u T u T u = I and thus z t + uu T u u T z t u T u T 3 x, y = { u 4i z t I Imz z t 4 z t + uu T u u T z t u T u Imz 4i z t u u T + uu T } z t u T. u In general, there holds z t = z t, t R, z C, which implies N k t P j s Imz z t ds dt z t N k t P j s Imz = ds dt z t = i Im N k t P j s Imz z t ds dt N k t P j s Imz z t ds dt 5.7 and N k t P j s Imz z t + ds dt z t N k t P j s Imz = ds dt + z t N k t P j s Imz = Re ds dt z t N k t P j s Imz z t ds dt. 5.8

62 56 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation Thus, we otain for the integration with 5.8, 5.9, 5.7 and 5.8 l P j x T 3 x, y T Ñ m k y ds y ds x Γ l Γ m = v { N k t P j s Im ds dt I z t Re Im N k t P j s Imz z t ds dt N k t P j s Imz z t ds dt uu T u u T u T u u u T + uu T u T u Applying the definition of Oj,k a, and Õ j,k a, completes the proof. Summarizing the results we have the following formula for the computation of the Galerkin entries for the doule layer operator. Theorem 5.3. Let Γ l, Γ m T h e oundary elements with Γ l Γ m =. Moreover, let u and v e the directional vectors of Γ l and Γ m and zs = a+ s as defined in Section 5.. Then, there holds for j N and k N K l,m j,k = v π Im Oj,k a, µ v I + πλ + µ Re Oj,k a, I I { λ + µ v uu Re Õ T u u T πλ + µ j,k a, u T u u u +Im Õ T + uu T } j,k a, u T. u } The Hypersingular Operator Instead of deriving an analytical formula for the computation of the Galerkin matrix of the hypersingular operator, as it is already done in the previous sections, we proceed as in [5] and use the formulas for the single layer operator and the property v in Lemma 3..5, i.e. W ψ, ψ = V d ds ψ, d ds ψ ψ, ψ H Γ. 5. We calculate for a fixed comination of oundary elements Γ l and Γ m and for fixed polynomial degrees j j p l and k k p m the -matrix Ñ Ñ Ñ m l m V d k d ds, j ds V d k d ds, ds Ñ l j Ñ V d l d ds Ñ m, j ds V d d. k ds Ñ m, ds k Ñ l j Therey, the arc length derivative of Ñ l j d dsñl j x = d ds is given y Nj γ x = N j γ l l x t T γ l,

63 Section 5.4: A Posteriori Error Estimators 57 where t denotes the unit tangential vector of Γ l with t = u Section 5.. Then, we get with B l A l = u t T γ l = u ut u u T u = u. u and u as defined in Moreover, using the definition of the Loatto shape functions we otain for j 3 N jt = P j t, N t = P t and N t = P t which yields d ds Ñ l j x = u d ds Ñ l x = u P l j x, P l x and d ds Ñ l x = u P l x. Thus, we calculate for j, k 3 m P 4 V k u v V, P m k l P, j P l j m P V k V P m k, P l j. 5., P l Note that the other cases j 3 or k 3 arise analogously. As we already stated in the introduction the operators V and V are only distinguished from the factors of the kernel function. Hence, we use the result from Lemma and Lemma and otain a formula for the calculation of the entries of the Galerkin matrix of the hypersingular operator. j 5.4 A Posteriori Error Estimators In this section we introduce error estimators for the Dirichlet, the Neumann and the mixed prolem. Since all estimators are provided y the epsbem framework we do not go into detail on the calculation and the implementation The Dirichlet Prolem and Symm s Integral Equation There are three different types of error estimator that are considered within this work for solving Symm s integral equation: i Error estimators that are created y space enrichment with respect to the mesh-size h - h/ estimators. ii Error estimators that are created y space enrichment with respect to the polynomial degree p - p estimators. iii Residual ased error estimators.

64 58 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation In order to define the error estimators we introduce the following notations. Let φ H Γ e the exact solution of Symm s integral equation 3.6, φ p h S T h, p e the numerical solution on a given triangulation T h and φ p h/ S T h/, p e the numerical solution on the uniformly refined triangulation T h/. Moreover, we denote y Π p h the L -projection on S T h, p and y the energy norm on Γ which is induced y the single layer operator, i.e. φ := V φ, φ. Definition 5.4. h-h/ error estimators The h-h/ error estimators for Symm s integral equation are given y i η h := φ p h/ φp h 5. ii η h := Π p h φp h/ 5.3 iii µ h := h / p + φp h/ φp h 5.4 L Γ iv µ h := h / p + Πp h φp h/ 5.5 L Γ. Definition 5.4. p-p error estimators For p p, the p-p error estimators for Symm s integral equation are given y i η p := φ p h φp h 5.6 ii η p := Π p h φp h 5.7 iii µ p := h / L p + φp h φp h 5.8 Γ iv µ p := h / p + Πp h φp h 5.9 L Γ. Definition Residual ased error estimator The residual ased error estimator for Symm s integral equation is given y ν := h / d p + ds V φp h f 5.3 L Γ, where d ds denotes the arc length derivative on Γ. Note that the η estimators cannot e used to create adaptive algorithms, since the energy norm cannot e localized meaning we cannot estimate the error on one oundary element. However, the µ estimators can e calculated locally and are consequentially appropriate for controlling the refinement in adaptive algorithms. In [8] the efficiency and reliaility of the h-h/ estimators for p = is proven, whereas there is no proof for the other error estimators. A detailed description of the calculations of all aove defined estimators can e found in [].

65 Section 5.4: A Posteriori Error Estimators The Neumann Prolem and the Hypersingular Integral Equation For the hypersingular integral equation we introduce two different types of error estimators, namely the h-h/ estimators and p-p estimators for p p. In order to defining the error estimators we introduce the following notation. Let u H Γ e the exact solution of the hypersingular integral equation 3.3, u p h S T h, p e the numerical solution on a given triangulation T h and u p h/ S T h/, p e the numerical solution on the uniformly refined triangulation T h/. Moreover, we denote y Π p h the L -projection on S T h, p and y the energy norm on Γ that is induced y the hypersingular operator, i.e. u := W u, u If Γ is connected we define u := u, u W +S. Definition h-h/ error estimators The h-h/ error estimators for the hypersingular integral equation are given y i η h := u p h/ up h 5.3 ii η h := Π p h up h/ 5.3 iii µ h := h / d p + ds up h/ up h 5.33 L Γ iv µ h := h / d Π p h p + ds h/ up L Γ Definition p-p error estimators For p p the p-p error estimators for the hypersingular integral equation are given y i η p := u p h up h 5.35 ii η p := Π p h up h 5.36 iii µ p := h / d L p + ds up h up h 5.37 Γ iv µ p := h / d Π p p h + ds up h L Γ Again, only the µ estimators are suitale for implementing adaptive algorithms. The efficiency and the reliaility of the h-h/ estimators for p = is proven in [7], whereas there is no proof for the other error estimator The Mixed Prolem For the mixed prolem we introduce the h-h/ and p-p estimators p p. In order to define the error estimators we introduce the following notation. Let η D,h, η D,h, η D,p, η D,p and µ D,h, µ D,h, µ D,p and µ D,p e the error estimators for Symm s

66 6 Chapter 5: Realization of the hp-bem for the Navier-Lamé Equation integral equation restricted to the Dirichlet oundary. Moreover, let η N,h, η N,h, η N,p, η N,p and µ N,h, µ N,h, µ N,p and µ N,p e the error estimators for the hypersingular integral equation restricted to the Neumann oundary. Definition h-h/ error estimators The h-h/ error estimators for the mixed prolem are given y i η h := η D,h + η N,h 5.39 ii η h := η D,h + η N,h 5.4 iii µ h := µ D,h + µ N,h 5.4 iv µ h := µ D,h + µ N,h. 5.4 Definition p-p error estimators For p p the p-p error estimators for the hypersingular integral equation are given y i η p := η D,p + η N,p 5.43 ii η p := η D,p + η N,p 5.44 iii µ p := µ D,p + µ N,p 5.45 iv µ p := µ D,p + µ N,p This definition implies that the results concerning the efficiency and the reliaility of the estimators can e transferred from the previous two susections.

67 Chapter 6 Implementation In this chapter we discuss the implementation. Since it is one of the main goals of this thesis to integrate all functions for solving the Navier-Lamé equation into the epsbem framework, we give a short overview on the software package in the first section. In the second section we go into detail on the implementation of the doule integrals that we introduced in Section 4. and finally, we descrie the implementation of the hp-bem for the Navier-Lamé equation. 6. Overview on the epsbem Package epsbem efficient and p-stale Boundary Element Methods is a software package to solve the Laplace and the Navier-Lamé equations with the hp-bem. Figure 6. shows an overview on all routines of the software package. The focus of this software package is on the efficient and stale implementation such that the results are accurate close to machine precision. The main routines are supposed to e implemented in Matla, such that an easy handling for the user is provided. Two C-liraries to calculate the associated Legendre functions and integrals thereof ottom row in Figure 6. are the core of the software package. Using openmp and multi-precision liraries these C-liraries provide an efficient and stale calculation of the asic integrals. Based on the C-liraries, the routines for the calculation of the Galerkin matrices are implemented in oth C and Matla. On the one hand the code is supposed to e understandale and short Matla-routines, on the other hand the routines are supposed to e efficient and stale C-routines. Additionally, there are routines for assemling and solving the linear system of equations, routines for displaying the solution and routines for refining the mesh using different mesh refining strategies. On the top layer, the error estimators and routines for calculating the exact error are implemented. Within the scope of this thesis, we add the C-functions O, U and Ut to the doule integral lirary, implement the Matla- and C-routines for the the calculation of the Galerkin matrices and adjust some plot routines for displaying the displacement field. Moreover, we investigate the incorporation of gliding conditions susequently defined for the Navier-Lamé prolem.

68 6 Chapter 6: Implementation Error Calculation errdirichlet estslpmu estslpmu Error Estimators esthypmu esthypmup oscdirichlet estslpmutilde estslpmutildep esthypmutilde esthypmutildep errneumann estslpeta estslpetap esthypeta esthypetap oscneumann estslpetatilde estslpetatildep esthypetatilde esthypetatildep /li/general estslpnu prolongationhh projectionhh /li/general Solution of IE SolveSymm_f uildsymmrhs_f SolveSymm uildsymmrhs SolveHypsing_f uildhypsingrhs_f Mesh Functions re neuniform re neadaptive hpmeshslit Display plotarclenthp plotarclenthn showsoldom Miscellaneous checkp checkp gauss SolveHypsing uildhypsingrhs hpmeshlshape showpot diagscaling SolveMixed uildmixedrhs uildhypsingstailization /li/general uildsortedmesh markelements /li/general showcomplex showboundarygrid /li/general loglogtriangle /li/general Potentials potv potk pota potw /li/lame/mat /li/lapace/mat Ansatz Spaces Legendre projectionl Loatto projectionh /li/general Galerkin Matla uildv uildw uildk uildm /li/lame/mat /li/lapace/mat Galerkin C uildv uildw uildk /li/lame/c /li/lapace/c Single Integrals Matla qtm qtm qt qt r r /li/general/mat Single Integrals C qtm qtm qt qt r r qtm_mpc qtm_mpc qt_mpc qt_mpc r_mpc r_mpc /li/general/c Doule Integrals Im I I Ut U O /li/general/c Fig. 6.: Overview on the functions of the epsbem package, where C-routines are depicted red and Matla-routines are depicted lue. Source:[]. 6. Integrals of the Associated Legendre Functions The functions for the calculation of the doule integrals Ij,k m and Om j,k are implemented according to formulas that are derived in [9] and - in the case of Oj,k - with the formulas that are proven in Section 4., respectively. Note that there are no routines for the calculation of the integrals Ĩ j,k and Õ j,k since these integrals are computed directly in the routines for the calculation of the Galerkin matrices using 4.6, 4.7, 4.8 and 4.9. The routines for the calculation of I j,k, I j,k, O j,k and O j,k are implemented in the C- lirary lilegendre.c and connected to Matla via mex-files. Therey, the doule integrals are calculated via recurrence relations in order to get an efficient implementation for high polynomial degrees p with complexity Op j =,.., p, k =,.., p +. Since the recurrence relations are not stale we use the multi-precision liraries mpfr The Multiple Precision Floating-Point Reliale Lirary, see [9] and mpc The Multiple Precision Complex Lirary, see [6] in order to ensure results that are accurate close to machine precision, i.e. the tolerance 4 is used. Using the liraries mpfr and mpc the numer of digits that are used for the calculations is computed heuristically with the function getdigits that is provided y the epsbem framework. In the following, we exemplarily descrie the implementation of Oj,k, as this function is one of the functions that is implemented within the scope of this thesis.

69 Section 6.: Integrals of the Associated Legendre Functions 63 mex-function Input: doule a, int p Output: doule* A matrix O a,,p C-functions Input: doule a_re, a_im, _re, _im int p, ndigits doule* Are, Aim matrices Output: - O_int a_re, a_im, _re, _im, p, ndigits, Are, Aim Input: mpc_t a, int p, ndigits doule* Are, Aim Output: - O a,, p, ndigits, Are, Aim Fig. 6.: Overview on the functions for the calculation of Oj,k a,. An overview on the routines for the calculation of Oj,k is given in Figure 6.. We see that the calculations are performed in two different levels red oxes. Therey, the first level is the interface etween Matla and C and the second level converts the input parameters that are given in doule precision to mpc variales and performs the computations. In the following we descrie the implementation of the two levels more closely. The first level is implemented y the mex-function O. The input parameters are the complex numers a, in doule precision and the polynomial degree p, and the output is the matrix A C p+ p+ which is given y O,a, O,p+a, A = Op,a, Op,p+a, The mex-function computes the numer of digits that are needed for the calculation, allocates the memory for the output matrix A and calls the C-function O int. Listing 6.: Excerpt of lilegendre.c: O int. void O_ int doule a_re, doule a_im, doule _re, doule _im, int p, int ndigits, doule * Are, doule * Aim 3 { 4 mpc_t a, ;

70 64 Chapter 6: Implementation 5 mpc_ init a, ndigits ; 6 mpc_ init, ndigits ; 7 /* set a and */ 8 mpc_ set_ d_ d a, a_re, a_im, MPC_ RNDNN ; 9 mpc_ set_ d_ d, _re, _im, MPC_ RNDNN ; /* call O */ Oa,,p, ndigits, Are, Aim ; /* clear a and */ 3 mpc_clear a; 4 mpc_clear ; 5 } The second layer is implemented y the functions O int and O. Listing 6. shows the code of O int. In the lines 4-6 the mpc variales a and are first defined and initialized, then the function O, which performs the calculations, is called line. Finally, a and are deleted lines 3-4. The computation of the matrix A is performed in the C-function O according to 4.3 with the formulas that are derived in Lemmas 4..4, 4..5 and Therefore, we do not go into detail on the complete implementation, here. However, we Q k descrie the computation of η and η a ± and take a closer look at the implementation for a ± = ± in the following. Listing 6.: Excerpt of lilegendre.c: O computation of η. eta =; if a_im * a_im - _im * _im < && 3 a_re * _im - a_im * _re /_im <= -+e -4 4 { 5 eta = -; 6 } else { 7 int cond, cond ; 8 cond = fas a_im + _im < e -4; 9 cond = fas a_im - _im < e -4; if cond && cond && a_re - _re <= -+e -4 cond && a_re + _re <= -+e -4 cond && a_re - _re <= -+e -4 3 { 4 eta = -; 5 } 6 } Listing 6. shows the calculation of η. The four situations in which conv{a +, a } {x R : x < } and thus η = are illustrated in Figure 6.3. In line -3 in Listing 6. we check the first situation. Therey, we check if conv{a+, a }

71 Section 6.: Integrals of the Associated Legendre Functions 65 a+! a+! a-! - - a-! a-! a+! a+! a-! - - Fig. 6.3: Configurations with η =. intersects the real axis Ima + Ima <, line and if the intersection is smaller than line 3. The remaining three situations are treated in line -. We check if a + R with a + line, if a R with a line or if a ± R with a ± line. As already mentioned aove, the tolerance 4 is used for the case differentiation. Listing 6.3: Excerpt of lilegendre.c: O calculation of Q k η a ±. mpc_add z, a,, MPC_RNDNN ; /* z = a+;*/ mpc_su z, a,, MPC_RNDNN ; /* z = a-;*/ 3 if eta == - { 4 mpc_neg z, z, MPC_RNDNN ; 5 mpc_neg z, z, MPC_RNDNN ; 6 } 7 qtm_ex z, p+, ndigits, c5; 8 qtm_ex z, p+, ndigits, c6; 9 if eta == for i =; i <=p+; i ++ mpc_su c5[i], c5[i], c6[i], MPC_RNDNN ; 3 else { /* multiply with -^k */ 4 for i =; i <=p+; i += /* even indices : c5 -c6 */ 5 mpc_su c5[i], c5[i], c6[i], MPC_RNDNN ;

72 66 Chapter 6: Implementation 6 for i =; i <=p+; i += /* odd indices : c6 -c5 */ 7 mpc_su c5[i], c6[i], c5[i], MPC_RNDNN ; 8 } Listing 6.3 shows the calculation of Q k η a + Q k η a. In line - 6 we compute z = a + and z = a and, if η =, z = a and z = a +, respectively. Then, we call the function qtm ex that computes Q k z, k =,..., p +, and save the result in the mpc arrays c5 and c6. In the case that η = we compute Q k η [ a + Q k η ] a line 9-, whereas we calculate k Q k η a + Q k η a in the case that η = line 3-8. We change the order in the sutraction for all odd indices k line 6-7. Hence, we add implicitly 4π i in the case of k =. Note that the implementation of ± a η and Q k η is done analogously. Finally, we descrie the implementation in the special case that a ± = ±, where we only refer to the case a + =. As we already stated in the proof of Theorem 4..7, the coefficients of the singular terms in the formulas for the computation of Õj,k a, vanish. Therefore, we set Q k to its finite part, i.e. Q k := W k, where we again use the tolerance of 4. We want to stress that this method only works if either only Q k a + or Q a k occur in the formula for the calculation of Õj,k a,. This effect is due to the fact that the sequences a n + n n N and an n n N converge to from different directions. Thus, comining Q k a + and Q a k results in a different value for a + =, which yields a discontinuous ehavior of the doule integral Õj,k a, in the neighorhood of. Note that the formulas that we derived for the computation of Õ j,k fulfill this constraint, whereas the formulas that are stated in [9] do not fulfill this constraint and cannot e used for the implementation. 6.3 Implementation of the hp-bem In this section we descrie the implementation of the hp-bem for the Dirichlet, the Neumann and the mixed prolems. Since the main routines and the routines for solving the three different prolems are provided y the epsbem framework and were only adjusted for the Navier-Lamé equation, we only descrie the procedure in solving the prolems ut not the implementation. However, in the susequent sections we go into detail on the implementation of the routines that are different from the routines for the Laplace equation and implemented within this work. As we already derived in Chapter 3 for solving the mixed, the Dirichlet and the Neumann prolems we solve Ax = g M A, 6. u D

73 Section 6.3: Implementation of the hp-bem 67 Vx = K + M u D 6. and W + Su = M KT g. 6.3 We proceed as follows: i We project the Dirichlet data that is given y the function handle u D and the Neumann data that is given y the function handle g to the discrete spaces S T h, p and S T h, p, respectively. For this, we use the Matla functions projectionh.m and projectionl.m and as a result we otain the coefficient vectors u D with respect to asis of S T h, p and g with respect to asis of S T h, p. ii Depending on the prolem we assemle the Galerkin matrices V, K and W y calling the functions uildv.m, uildk.m and uildw.m, the stailization matrix y calling uildhypsingstailization.m and the mass matrix y calling uildm.m. In the case of the mixed prolem we restrict the matrices V, K and W to the degrees of freedom of the Neumann and the Dirichlet oundaries, since 6. holds on Γ N Γ D. Thus, we solve a NN + N D N N + N D system. iii We solve the linear system of equations 6. with the ackslash operator of Matla and otain the coefficient vector of the numerical solution with respect to the ases of S T h, p and S T h, p. In the susequent sections we amplify the implementation of the Galerkin matrices and investigate another type of oundary conditions, namely gliding conditions Implementation of the Galerkin Matrices For the calculation of the Galerkin matrices there are oth C-functions and Matla functions. We descrie the implementation of the C- and the Matla functions y taking the example of the single layer operator. Note that the complete code for calculating the Galerkin matrices can e found in Appendix B. An overview on the Matla functions for the calculation of the Galerkin entries is given in Figure 6.4. All functions in the red oxes are implemented in the Matla file uildv.m. Figure 6.4 shows that the calculation is implemented on three different levels red oxes. In the lowest level we perform the calculation of the lock matrix that we introduced in Section 5.3. for one fixed comination of oundary elements. As we already stated in Section 5.3 we distinguish the case of identical elements galv id and non-identical elements galv ex. In the following we descrie the implementation of galv ex. The input parameters of galv ex are

74 68 Chapter 6: Implementation uildv uildv_ex uildv_quad galv_ex galv_id potv Fig. 6.4: Overview on the functions for the calculation of the Galerkin matrix for the single layer operator. vert x R : Matrix that contains the edges of Γ l, i.e. A l corresponds to the first row and B l corresponds to the second row. vert y R : Matrix that contains the edges of Γ m, i.e. A m corresponds to the first row and B m corresponds to the second row. p N : Polynomial degree. options: Struct that contains the Lamé coefficients λ and µ. The output val is the lock matrix that is introduced in Section Listing 6.4 shows an excerpt of the code. Listing 6.4: Excerpt of uildv.m: galv ex. % *** compute u,v,w,a,... 3 % *** compute matrices 4 uut = u *u / utu ; 5 usust = [u ;-u ]*[ u,-u ] / utu ; 6 uust = u *[ u,-u ] / utu ; 7 usut = [u ;-u ]*u / utu ; 8 % *** compute integrals Im 9 im = real Im a,,p; tmpm = [im, zeros p + ; zeros p +,im ]; % *** compute It = > comination of rows! i = [ zeros,p + ;Ia,,p + ];

75 Section 6.3: Implementation of the hp-bem 69 3 c = repmat : p +./[,3::* p + ],p +, ; 4 c = repmat : p./[,3::* p + ],p +, ; 5 tmp = a*i : end -,: p * c.* i 3: end,: p + +c.* i : end -,: p + ; 7 % *** compute matrices multiplied with intgrals 8 U = [ uut, -usust, * imag tmp,... 9 uut, -usust, * imag tmp ;... uut, -usust, * imag tmp,... uut, -usust, * imag tmp ]; U = [ uust, + usut, * real tmp,... 3 uust, + usut, * real tmp ;... 4 uust, + usut, * real tmp,... 5 uust, + usut, * real tmp ]; 6 % *** compute single layer potential 7 val =- fac * tmpm - fac *U -U; 8 % *** add deltak * deltaj 9 tmp = fac ** log utu * eye -* fac * uut ; 3 val [,p+],[, p +] = val [,p+],[, p +] - tmp ; After having computed the vectors u, v and w according to Figure 5. and a, C according to 5.3 we compute the matrices uu T u T u, T u u u T u and uu T + u u T u T u in line 4-7. In line 9- we compute the real part of the integral I j,k a, j,k=,...,p y calling the C-function Im and assemle the lock matrix tmpm = Re I j,k a, j,k=,..,p Re I j,k a, j,k=,...,p. Afterwards, we compute the matrix Ĩj,k a, in lines -6. Therefore, we j,k=,...,p call the C-function I to calculate the matrix Ij,k a, and comine the j,k=,...,p rows according to 4.6 and 4.7. Note that defining I,k a, := we can treat 4.7 similar to 4.6 for j = and thus the first row of i is initialized with zeros line. Since Matla can perform matrix operations efficiently, we comine all rows at once without a loop. We uild the coefficient matrices c := c j,k j,k=,...,p and c := c j,k j,k=,...,p with c j,k = j j + and c j,k = j + j +

76 7 Chapter 6: Implementation and comine the whole su matrices with the elementwise multiplication line 5-6. In line 8-4 we compute the lock matrices uu Im Ĩ j,k a, T u u T uu Im Ĩ j,k=,..,p u T u j,k a, T u u T, j,k=,..,p u T u, U = Im Ĩ j,k a, j,k=,..,p uu T u u T u T u, Im Ĩ j,k a, j,k=,..,p uu T u u T u T u, and U = Re Ĩ j,k a, uu T +u u T j,k=,..,p u T u, Re Ĩ j,k a, uu T +u u T j,k=,..,p u T u, Re Ĩ j,k a, uu T +u u T j,k=,..,p u T u, Re Ĩ j,k a, uu T +u u T j,k=,..,p u T u,. Finally, we compute Galerkin matrix according to 5., where we add the term with δ k, δ j, in lines 8-9. Note that in galv ex we distinguish the case that Γ l < Γ m in which we proceed as descried aove and the case that Γ l Γ m in which we change the order of the integration, as the formulas for the calculation of the numer of digits for Im and I prescrie that the element of the outer integral is smaller. One can show that y changing the order of integration we get v u w w a a. Besides these changes, the calculation that is descried aove is exactly the same in the other case. In the second level the routines uildv ex and uildv quad are implemented. The input parameters of oth functions are given y: coordinates R Nc : Matrix that contains the vertices of the oundary Γ ; each row contains the x - and x -coordinate of one edge. N c denotes the numer of edges of the triangulation. elements R N el : Matrix that descries the oundary elements; each row contains the indices of the edges of one oundary element. p N N el : Vector that contains the polynomial degree of each oundary element. options: Struct that contains the Lamé parameters λ and µ. Both routines return the Galerkin matrix V of the single layer operator. However, the routines uild ex and uildv quad differ in the calculation of the entries of

77 Section 6.3: Implementation of the hp-bem 7 the Galerkin matrix. uildv ex calculates the entries analytically y calling the functions galv ex and galv id, whereas the entries for non-identical elements are computed semi-analytically in uildv quad, i.e. the inner integral in Γ l l P j x Ux, y Γ m m P k y ds y ds x is computed analytically with the function potv that is descried in [] and the outer integral is computed vice quadrature. Thus, we otain Γ l l m P j x Ux, y P k y ds y Γ } m {{} =potvx ds x Γ l N g ν= ω g ν P j x g ν potvx g ν, 6.4 where N g denotes the numer of Gauss nodes and ων g and x g ν denote the Gauss weights and points. Defining the matrix P = p ν,j j=,...,p ν=,...,n g with p ν,j = P j x g ν and the matrix W R Ng p+ that contains the Gauss weights we can compute 6.4 for all j =,..., p and ν =,.., N g efficiently y Γ l l m P j x Ux, y P k y ds y ds x = Γ l Γ m W. PT potv. Here,. denotes the elementwise product and denotes the matrix-matrix multiplication. The top level in Figure 6.4 is used for exception handling and calling the su-routines for the analytical and semi-analytical computations. The C-routines for the analytical computation of the Galerkin matrices are implemented in the C-lirary ligalerkinlame.c. The structure of the routines is similar the structure of the Matla functions. Since the recurrence relations for the calculation of the doule integral I j,k a, and I j,k a, are very sensitive concerning the input parameters a and, the multi-precision liraries mpfr and mpc are used. In order to reduce the computational times we only compute a and with the multi-precision liraries, the other calculations are performed in doule precision. Moreover, the routines for the calculation of the Galerkin matrices are parallelized with openmp so that every thread calculates the su-matrix for a fixed comination of oundary elements. For a detailed description of the parallelization we refer the [3]. An analysis of the computational times for the assemly of the Galerkin matrices with oth the C- and the Matla routines is given in Figure 6.5. The calculations were performed on a computer with the following setup: Processor: AMD Phenom II X6 9T Operating System: Uuntu.4 RAM: 6GB DDR3. Figure 6.5 shows the computational time for several different methods, i.e. a geometric hp-method with 66 degrees of freedom green, a uniform h-method with 5 elements and p = lue, a uniform hp-method with 3 elements and p = 3

78 7 Chapter 6: Implementation Scaled Computaional Time 3/ / /3 /4 /5 /6 hp, 66 DOF n E =5, p = n E =3, p =3 n E =8, p = Numer of Threads Scaled Computational Time 3/ / /3 /4 /5 /6 hp, 66 DOF n E =5, p = n E =3, p =3 n E =8, p = Numer of Threads Scaled Computational Time 3/ / /3 /4 /5 /6 hp, 66 DOF n E =5, p = n E =3, p =3 n E =8, p = Numer of Threads Fig. 6.5: Computational times for the assemly of the Galerkin matrices V top, K middle and W ottom with the C-functions depending on the numer of threads. The lack dashed line is the optimal scaling and the horizontal lines denote the computational time for the assemly with Matla routines.

79 Section 6.3: Implementation of the hp-bem 73 dashed light lue and a uniform p-method with 8 elements and p = 7 red. The times that are illustrated in the plots are the average times of three runs each and scaled with the computational time of the C-routines with one thread. The results that we otain for the Navier-Lamé equation are similar to the results that are descried in [] for the Laplace equation. All three plots show that we do not achieve the optimal scaling lack dashed line for the computations with the C-routines, the maximal scaling factor that is achieved using 6 threads is in the range of This is what we expected, since every thread calculates a fixed numer of locks of the Galerkin matrices, i.e. the locks are not distriuted dynamically to the threads. Thus, if the numer of locks is not divisile y the numer of threads, some threads are finished while other threads calculate an extra lock. Additionally, the threads lock each other while writing the results into the Galerkin matrix, since the entries for the hat function are added up and thus exclusive writing is necessary. Moreover, we see that the scaling factor for V is worse than the scaling factors for K and W. This is due to the fact that the computational effort for computing a matrix lock is higher for the doule layer and the hypersingular operator than for the single layer operator and hence the parallelization has a igger impact. Comparing the Matla routines to the C-routines with one thread we see that the Matla routines are slower than the C-routines except for the calculations of V with high polynomial degrees. This is due to the fact that C can perform simple calculations very efficiently, whereas Matla can perform matrix operations very efficiently. Thus, the Matla routines for the calculation of V are faster than the C-routines for high polynomials degrees, since we only used matrix operations in galv ex. For h-methods the matrix locks are smaller and the matrix operations have less impact and consequentially the Matla functions are slower. Furthermore, since the computational effort for computing K and W is igger than for V the efficient matrix operations of Matla have a smaller impact and thus, the Matla routines for the doule layer and the hypersingular operator are slower than the C-routines even with high polynomial degrees Mixed Prolem with Gliding Conditions In this section, we discuss the implementation of another type of oundary conditions that occurs in many applications in the field of linear elasticity, namely the gliding conditions. Besides the Dirichlet oundary Γ D where the displacement is given and the Neumann oundary Γ N where the traction is given, we denote the gliding oundary y Γ G and prescrie the gliding conditions, i.e. the displacement of the ody is fixed in one specified direction n R and the traction is fixed in another specified direction m R. Thus, we get for h H Γ G H Γ G the following condition n T γ u m T = h on Γ γ u G. 6.5 We do not go into detail on the theory of the mixed prolems with gliding conditions, i.e. the existence and the uniqueness of a solution, ut discuss the numerical realization.

80 74 Chapter 6: Implementation A standard example in linear elasticity that can e solved using gliding conditions is the memrane with a hole that is illustrated in Figure 6.6. Fig. 6.6: Memrane with a hole where a surface force is applied on the upper and lower oundary left, part of the memrane with gliding conditions displayed y the circles right. As we can see in Figure 6.6 left there holds Γ = Γ N with γ u = gx :=, T on the lower oundary, γ u = gx :=, T on the upper oundary and γ u = gx :=, T on the left and right oundary. Instead of solving this Neumann prolem, we split the memrane in four parts and only calculate the solution on one of the parts exploiting the symmetry of the solution see Figure 6.6, right. Therey, we get gliding conditions on the cut edges, i.e. there holds γ u = γ u on the left oundary and γ u = γ u on the lower oundary. For incorporating the gliding conditions, we cannot proceed as we did for the mixed prolem in Section 3., to e more precisely, we cannot use the symmetric formulation of the Caldéron system 3.7. Instead we rearrange the Caldéron system 3.6 as follows V I + K I K W γ u = γ u For the implementation we prescrie a sorted mesh, i.e. dirichlet elements = neumann gliding.

81 Section 6.3: Implementation of the hp-bem 75 and we denote the coefficient vector of u p h y u = u D, u N, u G, u D, u N, u G and the coefficient vector of φ p h y T x = x D, x N, x G, x D, x N, x G. Using the previously defined Galerkin matrices we otain the discrete system V M K x M + =, KT W u Then, the oundary conditions are incorporated y extending the linear system of equations to V M K M T x M + KT T W M u =, 6.6 M M Λ f where Λ denotes the Lagrange parameter and the vector f = u D,, u D,, g, g, h, h T denotes the coefficient vector of the given Dirichlet, Neumann and gliding data. The matrices M and M are given y M = I I m I m I T and I I M =, n I n I where the upper part of oth matrices corresponds the Dirichlet data, the middle part to the Neumann data and the lower part to the gliding data. Thus, the given oundary conditions are fulfilled due to last equation of the extended system M x + M u = f. Note that the system 6.6 is not appropriate for solving the mixed prolem without gliding conditions as there are also disadvantages compared to the system of linear

82 76 Chapter 6: Implementation equations for the mixed prolem 6.. First, system 6.6 is not symmetric and thus, efficient algorithms, such as the cg-method, cannot e used for solving the system. Moreover, using 6.6 we have to solve a igger system of linear equations. Besides the extension of the Caldéron system y the matrices M and M we do not restrict the Galerkin matrices V, K and K to the degrees of freedom of the Dirichlet and the Neumann oundary as it is done in 6.. Therefore, the extended system is of the size N + N + N D + N N N + N + N D + N N. The assemly and the solving of the linear system of equations 6.6 is implemented in the Matla function solvemixedgliding.m. The input parameters are: coordinates R Nc : See input parameters uildv ex. dirichlet R Nel,D, neumann R N el,n, gliding R Nel,G : Matrix that descries the oundary elements on the Dirichlet, the Neumann and the gliding oundaries; each row contains the indices of the edges of one oundary element. N el,d, N el,n and N el,g denote the numer of elements on Γ D, Γ N and Γ G. u, g: Function handles of the Dirichlet and the Neumann data. m, n R N el,g : The directions in which the displacement and the traction are fixed for every gliding oundary. In the following we descrie the assemly of the matrices M and M more closely. Although the matrices M and M are sparse, we adicate the sparse format and initialize oth matrices as full matrices, since the Galerkin matrices V, K and W and thus the complete matrix in the system of linear equations are in general densely populated. In order to assemle the lower part of the matrices M and M, we first extend the matrices m and n to the size NG and N G and save the result in the variales ctmp and ctmp. The corresponding excerpt of the code of solvemixedgliding is presented in Listing 6.5. Note that the complete code can e found in Appendix C. freenodesg = unique gliding ; ctmp = []; ctmp = []; 3 for k =: length freenodesg Listing 6.5: Excerpt of solvemixedgliding.m. 4 [idx,~]= find gliding == freenodesg k; 5 ctmp = [ ctmp ; n idx,: ]; 6 end 7 for k =: size gliding, 8 % dof on k- th gliding oundary with respect to S 9 idx = [ cp ned + nen +k: cp ned + nen +k + -]+;

83 Section 6.3: Implementation of the hp-bem 77 % dof on k- th gliding oundary with respect to S idx = cp ned + nen +k - +: cp ned + nen +k; % extend m and n 3 ctmp = [ ctmp ; repmat mk,:,length idx, ]; 4 ctmp = [ ctmp ; repmat nk,:,length idx, ]; 5 end Extending m is implemented with a loop over all elements on Γ G. For extending n we need oth a loop over all nodes and a loop over all elements on Γ G, since in S T h, p there are oth nodal asis functions hat functions and asis functions of higher polynomial degree that live on one element. After having extracted the indices of all nodes on the gliding oundary line we loop over all nodes on Γ G and append the row of n that corresponds the the k-th node to ctmp line -6. The loop over all elements of the gliding oundary is implemented in lines 7-5. First, we extract the indices of the degrees of freedom on the k-th oundary element with respect to the asis of S T h, p idx and the asis of S T h, p idx in lines 8-. Using the Matla-function repmat we extend the rows of m and n that correspond to the k-th oundary element and append it to ctmp and ctmp, respectively. Finally, the entries for the gliding data in M are created y the Matla expression diagctmp:, and diagctmp:,, the entries in M y ctmpn:, and diagctmp:, and inscried into the matrices.

84

85 Chapter 7 Numerical Results In this chapter we present the numerical results. One focus is on the verification of the convergences rates and the efficiency and reliaility of the error estimators for several refinement strategies. As we already stated in the introduction the results have to e accurate close to machine precision. However, in the implementation we calculate the squared error estimators and the squared energy error and extract the square root afterwards, where we loose half the precision. Hence, we want to reduce the error and the error estimators y 7-8. In the first section we illustrate the results for Symm s integral equation and the Dirichlet prolem, and in the second section the results for the hypersingular integral equation and the Neumann prolem are presented. This chapter closes y showing the results for the mixed prolem with and without gliding conditions. 7. The Dirichlet Prolem Slit Example As a first example, we calculate V φ = on the slit, i.e. Γ := [, ]. For all computations on the slit we choose λ = 6 and µ = 3. Since there is no analytical solution for this prolem, we compute the energy norm of the exact solution φ y extrapolation in order to compute the error in the energy norm y φ φ p h Gal. Orth. = φ φ p h. In the following we verify the convergence rate that is given in 3.8. For a constant right-hand side f it can e shown that φ H ε Γ, ε >, and thus we expect a convergence rate of O [N ] for a uniform h-method and O [N ] for a uniform p-method. In Figure 7. the numerical results for uniform h- and p-refinements are illustrated. The plots show the error and the error estimators plotted against the degrees of freedom with a doule logarithmic scaling. Note that for all p p estimators p = p+ is used. We see that all lines run parallel with the slope of for the uniform h-method and for the uniform p-method, i.e. the convergence rates coincide with the expected convergence rates. Note that it can e shown numerically that increasing the polynomial degree for the uniform h-method or pre-refining the mesh for the uniform

86 8 Chapter 7: Numerical Results error and estimators N φ φ p h ν µ p µ p η p η p µ h µ h η h η h error and estimators N φ φ p h ν µ p µ p η p η p µ h µ h η h η h Fig. 7.: Error and error estimators for a uniform h-refinement with p = top and a uniform p-refinement with two elements ottom for V φ =, T on the slit. p-method does not yield a etter convergence rate, ut we otain a etter constant in 3.8. Figure 7. shows the error and the error estimators plotted against the degrees of freedom using an adaptive h-refinement with p = in the first plot and p = 4 in the second plot. Again, the plots are scaled doule logarithmically and p = p + is used for the p p estimators. Moreover, we mark the elements in the adaptive algorithms with the Dörfler criterion θ =.55 using the estimator µ h. We see that we get a convergence rate of O [N ] 3 for p = and O [N ] for p = 4, which corresponds to the optimal convergence rate of O [N ] p+ 3

87 Section 7.: The Dirichlet Prolem 8 error and estimators φ φ p h ν µ p µ p η p η p µ h µ h η h η h N error and estimators N φ φ p h ν µ p µ p η p η p µ h µ h η h η h Fig. 7.: Error and error estimators for an adaptive h-refinement with p = top and p = 4 ottom for V φ =, T on the slit. that can e reached according to 3.8. Thus, using an adaptive h-refinement the regularity of the solution does not spoil the convergence rate. Note that for p = 4 the slope of the estimator ν varies from in the last three steps. However, we already reduced the estimator y 7 digits and cannot assume that the results are accurate anymore. In Figure 7.3 we see the error and the error estimators plotted against the square root of the degrees of freedom for an adaptive hp-refinement in the first plot and a geometric hp-refinement in the second plot. Here, only the y-axis is scaled logarithmically. We use the error estimator µ h and the Dörfler criterion for oth marking the elements for mesh refinement θ h =.55 and for marking the elements for increas-

88 8 Chapter 7: Numerical Results error and estimators 4 φ φ p h ν µ p µ p η p η p µ h µ h η h η h 6 N error and estimators φ φ p h ν µ p µ p η p η p µ h µ h η h η h 5 N Fig. 7.3: Error and error estimators for an adaptive hp-refinement top and a geometric hp-refinement ottom for V φ =, T on the slit. ing the polynomial degree θ p =.8. According to [7] we expect an exponential convergence rate, to e more precisely, there holds φ φ p h e β N, β >. 7. Thus, we expect a straight line with slope β loge in the plots. Figure 7.3 shows that we otain an exponential convergence rate for oth the adaptive and the geometric hp-refinement. However, we see that we otain a etter constant for the adaptive algorithm than for the geometric algorithm, i.e. there hods β adap. > β geom.. This can e explained y the fact that we use a posteriori knowledge for the adaptive algorithm, whereas we prescrie the refinement a priori in the geometric algorithm. Additionally, we see that the slope of the residual ased estimator in the first plots varies in the last steps. Again, we already reduced the

89 Section 7.: The Dirichlet Prolem 83 estimator to machine precision and cannot assume the results to e accurate anymore. All figures for the slit example show that the error estimators run parallel to the energy error, which proofs the efficiency and the reliaility of these estimators numerically for the slit example. Rotated L-shape Example, -,,,, -,-,-,- Fig. 7.4: Rotated L-shape domain. The second example that is treated in this section is a common enchmark example on the rotated L-shape that is also investigated in [], [4] and [8]. The rotated L-shape is illustrated in Figure 7.4 and the exact solution u := u r, u ϕ T is given in polar coordinates y u r r, ϕ = rα { [ ] [ α + cos α + ϕ + C α + ] C cos [ α ϕ ]} µ u ϕ r, ϕ = rα { [ ] [ α + sin α + ϕ + C + α ] C sin [ α ϕ ]}, µ where the parameters are defined y C = cos[α+ω] cos[αω], C = λ+µ λ+µ with ω = 3 4 π. Moreover, α is the positive solution of α sinω + sinω α =. In order to compare the numerical results with [] we choose the parameters of material of ronze E = and ν =.3, which implies that λ and µ = In Figure 7.5 the h-h/ estimator µ h is plotted against the degrees of freedom for a uniform h and a uniform p-refinement, an adaptive h-refinement with p = 4 and an adaptive hp-refinement. For oth adaptive algorithms the Dörfler criterion with

90 84 Chapter 7: Numerical Results adaptive hp adaptive h, p =4 uniform p uniform h µh N Fig. 7.5: µ h for a uniform h- and p-refinement, an adaptive h-refinement with p = 4 and an adaptive hp-refinement. adaptive hp adaptive h, p =4 3 µh N Fig. 7.6: L-shape: µ h for an adaptive h-refinement with p = 4 and an adaptive hp-refinement with semi-logarithmical scaling. θ =.55 for the adaptive h-method, and θ h =.55 and θ p =.8 for the adaptive hp-method is used. Additionally, we mark the elements using the estimator µ h. Figure 7.5 shows the following results: Even with the adaptive algorithms the error estimator is only reduced y 6 digits. This is due to the fact that for approximating the given Dirichlet data we need a very high polynomial degree p, i.e. p = p + is chosen in

91 Section 7.: The Dirichlet Prolem 85 this example. Thus, we have to store ig matrices and need a large numer of digits for the computations with the multi-precision liraries. This results in very long computational times. The adaptive h-method converges faster than the adaptive hp-method. This can e explained y the fact that the parameters for marking the elements are etter chosen for the adaptive h-method than for the adaptive hp-method. However, Figure 7.6 shows, that we do not otain an exponential convergence rate for the adaptive h-method. The adaptive hp-method converges exponentially, although the convergence is slow. For all refinement strategies we otain the same convergence rates as for the slit example which indicates numerically that φ H ε Γ. Fig. 7.7: L-shape: Asolute value of displacement with displacement field on the domain Ω. Figure 7.7 shows the exact solution of the Dirichlet prolem plotted in the domain Ω, where the vector field denotes the displacement on a grid and the colors denote the asolute value of the displacement in every point of the grid. Figure 7.8 shows the oundary Γ red dashed and the deformed oundary Γ, i.e. Γ x := x + ux, x Γ. In order to visualize the displacement, it is scaled with the factor 3 as it is also done in [8] and []. Both figures coincide with the

92 86 Chapter 7: Numerical Results Fig. 7.8: L-shape: Undeformed oundary Γ red dashed and deformed oundary Γ lue scaled with factor 3. figures that are illustrated in [8] and [], although no exact values are presented, there. 7. The Neumann Prolem Slit Example As a first Neumann example we solve W u = on the slit. As we already did for solving Symm s integral equation, we prescrie the Lamé parameters λ = 6 and µ = 3 and determine the energy of the exact solution u y extrapolation since there is no analytical solution. Again, we verify the convergence rates that we stated in Section 3.3 See For a constant right-hand side it is known that u H ε Γ, ε >, and therefore 3.39 implies a convergence rate of O [N ] for a uniform h-refinement and a convergence rate of O [N ] for the uniform p-refinement. Figure 7.9 shows the errors and the estimator plotted against the degrees of freedom in a doule logarithmic scaling. We see that the convergence rate of the uniform h-method is O [N ] and the

93 Section 7.: The Neumann Prolem 87 convergence rate of uniform p-method is O [N ], which is what we expected. φ φ p h error and estimators 3 4 µ p µ p η p η p µ h µ h η h η h N error and estimators N φ φ p h µ p µ p η p η p µ h µ h η h η h Fig. 7.9: Error and error estimators for a uniform h-refinement with p = top and a uniform p-refinement with two elements ottom for W u =, T on the slit.

94 88 Chapter 7: Numerical Results Figure 7. shows the error and the error estimators plotted against the degrees of freedom for an adaptive h-method, where we used the Dörfler criterion with θ =.55 and µ h for refining the mesh adaptively. Referring to 3.39, the optimal convergence rate that we can otain y refining the mesh is O [N ] p+. Again, the numerical results coincide with the theory, since we get a numerical convergence rate of O [N ] 3 for p = and O [N ] 3 for p = 6. φ φ p h error and estimators µ p µ p η p η p µ h µ h η h η h N φ φ p h error and estimators N µ p µ p η p η p µ h µ h η h η h Fig. 7.: Error and error estimators for an adaptive h-refinement with p = top and p = 6 ottom for W u =, T on the slit. For the geometric hp-refinement it is proven in [7] that there holds an error estimation that is similar to 7. for the hypersinguar equation, i.e. we can expect an exponential convergence rate. The adaptive and geometric hp-methods are illustrated in Figure 7. semi-logarithmically plotted against the square root of the

95 Section 7.: The Neumann Prolem 89 degrees of freedom. We see that oth methods converge exponentially and the the constant in the estimation is etter for the adaptive hp-refinement. Thus, the results are similar to the results for Symm s integral equation. error and estimators φ φ p h µ p µ p η p η p µ h µ h η h η h N error and estimators φ φ p h µ p µ p η p η p µ h µ h η h η h N Fig. 7.: Error and error estimators for an adaptive hp-refinement top and a geometric hp-refinement ottom for W u =, T on the slit. Square Memrane Example As a second Neumann example we illustrate the square memrane that is given y Ω := [ 3, 3] and that is stretched on the upper and lower side, i.e. a traction g = n is applied, where n denotes the outer normal vector. The Lamé coefficients are given y λ = 6 and µ = 3. The results are illustrated in Figure 7. and 7.3. Note that we scaled the displacement with the factor for the visualization in Figure 7.3.

96 9 Chapter 7: Numerical Results Both Figures show that the memrane contracts in x -direction and expands in x -direction and the maximal displacement of is located at the edges. Moreover, we see that the displacement is symmetric to the center, i.e. this example can also e calculated y partitioning the domain in four parts using gliding condition. Fig. 7.: Square Memrane: Asolute value of displacement with displacement field on the domain Ω.

97 Section 7.3: The Mixed Traction and Displacement Prolem Fig. 7.3: Square Memrane: Undeformed oundary Γ red dashed and deformed oundary Γ lue scaled with factor. 7.3 The Mixed Traction and Displacement Prolem Cook s Memrane Example As a first example we choose a common example of linear elasticity that is also treated in [] and [8] and that corresponds to the two-dimensional model of the ar that we introduced in Section., namely the Cook s memrane example. The memrane that we investigate in the following is given in Figure 7.4. The memrane is fixed on the left-hand side, i.e. we have a Dirichlet oundary with u D x =, T and a shearing load is applied on the right-hand side, i.e. we have a Neumann oundary with gx =, T. Moreover, the top and the ottom oundary are also Neumann oundaries with gx =, T and volume forces vanish. Thus, we have a mixed prolem. According to [] we choose the parameter of material of Plexiglass, i.e. E = 9 and ν =.4, which implies that λ = and µ = The error estimator µ h is illustrated in Figure 7.5 for different refinement strategies. We see that we otain a exponential convergence rate for the adaptive hp-method and algeraic convergence rates with the slope of for the uniform h-method,

98 9 Chapter 7: Numerical Results 48,6,44 48,44, Fig. 7.4: Cook s memrane with Ω = conv{,, 48, 44, 48, 6,, 44} and a surface force applied on the right-hand side. µh adaptive h with p = adaptive hp uniform p uniform h 3 N N + N D Fig. 7.5: Cook s memrane: µ h for a uniform h- and p-refinement, an adaptive h-refinement with p = and an adaptive hp-refinement. for the uniform p-method and 3 for the adaptive h-method with polynomial degree p =. Using the adaptive hp-algorithm we reduce the estimator y 7 digits and thus we are accurate close to machine precision. Figure 7.6 and 7.7 show the displaced oundary and the displacement field, respectively. Note that we used a scaling factor of 5 for illustrating the displaced oundary. We see that the memrane ends up, whereas the left-hand side is fixed. The displacement proceeds continuously from the left to the right side, where the

99 Section 7.3: The Mixed Traction and Displacement Prolem 93 maximal displacement of.4 is located on the upper right edge of the memrane. The second Figure shows that the upper side of the memrane is ulged and the lower side is stretched. Thus, we see the typical displacement of a ar that is loaded as descried aove. Moreover, in [] and [8] similar result are illustrated, although no exact values for a comparison are given. Fig. 7.6: Cook s Memrane: Asolute value of displacement with displacement field on the domain Ω.

100 94 Chapter 7: Numerical Results Fig. 7.7: Cook s Memrane: Undeformed oundary Γ red dashed and deformed oundary Γ lue scaled with factor 5. Memrane with a Hole As a last example we investigate the memrane with a hole that we introduce in Figure 6.6. We solve the prolem according to the right figure in 6.6 using gliding conditions. Figure 7.8 shows the geometry with the initial mesh that is used for the calculations, where the gliding oundary is denoted y the circles. Again, we choose the parameters of material of Plexiglass. The directions in which the displacement and the traction is fixed on the gliding oundary are given y n =, T and m =, T for the left oundary and y n =, T and m =, T for the lower oundary. On the upper oundary we prescrie a traction gx :=, T, whereas we set gx =, T on all other Neumann elements.

101 Section 7.3: The Mixed Traction and Displacement Prolem 95,3 3,3,.5.5, 3, Fig. 7.8: Right upper part of the memrane with a hole with initial mesh red points and a traction on the upper side lue arrows. Fig. 7.9: Memrane with a hole: Asolute value of displacement with displacement field on the domain Ω.