Diese Arbeit wurde vorgelegt am Lehrstuhl für computergestützte Analyse technischer Systeme

Transcription

1 Diese Arbeit wurde vorgelegt am Lehrstuhl für computergestützte Analyse technischer Systeme Implementierung eines nichtlinearen isogeometrischen Schalenelementes für Fluid-Struktur-Interaktionen mit freien Oberflächen Implementation of a Nonlinear Isogeometric Shell Element for Fluid-Structure Interaction with Free-Surface Flows Seminararbeit Seminar Thesis von / presented by Spenke, Thomas Vollständiger Name des/r Prüfer/in / Dipl.-Ing. Norbert Hosters Aachen, 8. Januar 2017

2 Seminar Thesis Contents Glossary Indices Acronyms List of Figures I III III IV 1 Introduction 1 2 Underlying Theory Structural Mechanics The Structural Problem Isogeometric Analysis Isogeometric Reissner-Mindlin Shell Theory Computational Fluid Dynamics Navier-Stokes Equations Solution Strategies Fluid-Structure Interaction Coupling Conditions Partitioned Solution Algorithm Results Structural Mechanics Cantilever Beam Subjected to End Moment Channel Section Beam Subjected to Point Load Fluid-Structure Interaction Sloshing Tank Conclusion 20 References 21

3 Seminar Thesis I Glossary Symbols Structural Mechanics X x u u tt ρ T b F E S C Ñ t 0 b 0 δu δe M K U U Ü F R Description Position vector in the reference configuration Position vector in the current configuration Unknown displacement of the continuous problem 2nd time derivative of the displacement, i.e., acceleration Material density Cauchy stresses in the current configuration Vector of body forces in the current configuration Deformation gradient Green-Lagrange strain tensor 2nd Piola-Kichhoff stress tensor Material tensor, constant for a linear stress-strain relation Unit normal vector in the reference configuration Prescribed traction on Neumann boundary Vector of body forces in the reference configuration Virtual displacement Virtual strains Mass matrix Stiffness matrix Solution vector, i.e., discrete displacements Incremental displacements in Newton-Raphson iteration Discrete accelerations Right-hand side vector of external loads Vector of inner stresses (from previous solution steps) Isogeometric Analysis N p I (ξ) ξ 1, ξ 2, B I Ñ p i (ξ) w I Ξ ξ i n p NURBS basis functions Parametric NURBS coordinates Control points B-Spline basis functions Weights of control points Knot vector Knots Number of control points NURBS degree

4 Seminar Thesis II Symbols Description Isogeometric Shell Theory X(ξ α ), x(ξ α ) Position vectors of the shell midsurface X I, x I Control points of the shell midsurface X(ξ i ), x(ξ i ) Global position vectors of the structural body D(ξ α ), d(ξ α ) Director vectors D I, d I Nodal, i.e., control point associated, director vectors h Shell thickness G α Tangent vectors given by NURBS derivative A i (ξ α ) Local Cartesian coordinate system A ii, a ii Nodal local Cartesian coordinate systems θ i Coordinates of the local Cartesian system J αβ Jacobian relating local and parametric coordinates ε Green-Lagrange strain vector of the shell midsurface ε αβ Membrane strains κ αβ First-order curvatures (strains) ϱ αβ Second-order curvatures (strains) γ α Transverse shear strains σ 2nd Piola-Kirchhoff stress vector of the shell midsurface n αβ Membrane forces m αβ Bending moments q α Shear forces D Material matrix referring to the shell midsurface C P, C S Matrices modeling the translational and the rotational part of the linear-elastic isotropic material law µ G Determinant of the shifter tensor E Young s modulus ν Poisson ratio κ S Shear correction factor T 3I Transformation matrix from local to global coordinates R I Rotation tensor for rotational update ω I, ω I Axial vector of rotation and its increment β I Rotational solution increment v Displacement of the shell midsurface including rotations b0 Body forces acting on the shell midsurface B I Node-level linear strain-displacement matrix G, G IK (Nodal) initial stress matrix M e IK Node-level mass matrix F e I = F e I,ext Nodal right-hand side vector of external loads R e I = F e I,int Nodal vector of inner stresses (from previous iterations)

5 Seminar Thesis III Indices Symbols Description α, β Take the values 1, 2 in index notation i, j Take the values 1, 2, 3 in index notation I, K Refer to the associated control points Acronyms Abbreviation ACM CAD CAE CATS CFD EMUM FEAFA FSI NURBS ODE Description Aeroelastic Coupling Module Computer-Aided Design Computer-Aided Engineering Chair for Computational Analysis of Technical Systems Computational Fluid Dynamics Elastic Mesh Update Method Finite-Element Analysis For Aeroelasticity Fluid-Structure Interaction Non-Uniform Rational B-Spline Ordinary Differential Equation

6 Seminar Thesis IV List of Figures 1 Cantilever Beam: Setup Cantilever Beam: Graph Cantilever Beam: Rolled Up Channel Section Beam: Setup Channel Section Beam: Structural Response Channel Section Beam: Convergence Graph Sloshing Tank: Setup

7 Seminar Thesis 1 1 Introduction The persistent technological progress increases the demand for cost-efficient and lightweight structural components in numerous branches of industry, e.g., civil and aeronautical engineering, without making compromises in terms of stability and reliability. Shell structures are thin-walled structures that live up to these requirements by limiting stability to their main load axes: While shells are capable of carrying huge transversal stresses, they show a rather high sensitivity to loads applied in thickness direction[5]. In order to run numerical simulations of shell structures, mathematical shell models are required. Typically, these shell elements represent the structure as a reference surface plus some interpolation terms in thickness direction[1]. The shell element implemented in this thesis is based on isogeometric analysis (IGA), a discretization concept introduced by Hughes et al.[7] that is aimed at geometrical exactness as well as a stronger linkage between design and analysis. Due to the increasing expectations on numerical simulations in terms of accuracy the range of applications in which effects of fluid-structure interaction (FSI) have to be considered, e.g., aircraft wings or tank walls, is ever-expanding. Therefore, solution strategies for FSI problems are among the central research topics of the Chair for Computational Analysis of Technical Systems (CATS) at RWTH Aachen University. The primary goal of this seminar thesis has been the implementation and validation of a geometrically nonlinear isogeometric shell element capable of predicting the structural response of thin-walled structures for loading cases involving large deformations, large rotations, and finite strains. The major steps on this way included: The acquisition of expertise in the field of nonlinear shell theory, especially in the context of isogeometric analysis. An investigation of nonlinear isogeometric shell elements published in literature with regard to a possible implementation into the structural solver FEAFA (Finite-Element Analysis For Aeroelasticity) developed at CATS RWTH. Here, the element presented by W. Dornisch in his dissertation[9] has turned out to be a suitable basis. The modulation and implementation of the shell element as well as its validation in structural benchmark tests. Furthermore, the application of the new shell element to numerical simulations of fluid-structure interaction was intended. The underlying theory is introduced in Sec.2, in particular focusing on isogeometric shell theory, Sec Afterwards, the employed test cases and the corresponding numerical results are discussed in Sec.3, before this seminar thesis ends with a conclusion of which goals have been achieved and an outlook on possible future steps in Sec.4.

8 Seminar Thesis 2 2 Underlying Theory 2.1 Structural Mechanics The Structural Problem In general, the aim of structural mechanics is to calculate the deformation of a structural body caused by external loads or stresses. Typically, the solution is expressed as the displacement, which is the difference in the position of a material point between the deformed and the undeformed configuration, i.e., u = x X. The governing equation of the general structural problem, the equation of motion, is derived from the momentum balance and reads[18] ρ u tt = div(t ) + ρ b. (1) Here, ρ is the (constant) material density, while u denotes the displacement and, as a consequence, u tt the acceleration. The inner Cauchy stresses are represented by T, whereas b stands for the body forces acting on the structure, e.g., gravity. Obviously, Eqn.(1) is not closed, and therefore additional relations are required in order to solve the structural problem. These constitutive equations strongly depend on the underlying modeling assumptions made for the behavior of the body and its material. A common categorization is the distinction between linear and nonlinear models: Linear structural models assume a linear relation between external loads and resulting structural deformation[1]. Clearly, this rigorous assumption can only be justified for small deviations from the undeformed geometry. In contrast, structural models are called nonlinear whenever this postulation is omitted. There are two main aspects of nonlinearity that might be considered: Geometrically nonlinear formulations take into account that a structural deformation in general comes along with inner stresses which, vice versa, interfere with the deformation being in progress[1]. The usage of a nonlinear material law, i.e., stress-strain relation, for example by incorporating inelastic effects, is denominated by physical nonlinearity[1]. The structural model employed throughout this seminar thesis is geometrically nonlinear, but physically linear. As a result, it is capable of adequately representing large deformations, large rotations, but only small strains. The equilibrium formulated in Eqn.(1) refers to the unknown deformed configuration. However, for nonlinear problems it has to be linearized by formulating the constitutive laws based on a known geometry. In this thesis, a total Lagrangian description is

9 Seminar Thesis 3 chosen, so that all geometrical quantities are mapped back to the undeformed reference configuration with the aid of the deformation gradient F = dx/dx, that establishes a mapping between the reference and the current configuration[1]. Therefore, the Green-Lagrange strain tensor E is used as the (nonlinear) kinematic relation, i.e.,[9] E = 1 2 ( F T F I ), (2) with the identity matrix I. It is a measure for the strains in the current configuration with respect to the undeformed geometry. The stress measure energy conjugate to the Green-Lagrange strains is the 2nd Piola-Kirchhoff stress tensor S. As physical linearity is assumed, the two quantities are related by a linear stress-strain relation. It reads[1] S = C : E, (3) where C denotes the constant material tensor. Beyond that, boundary conditions play an important role in structural mechanics. There are two types of boundary conditions commonly used[9]: A Dirichlet condition u = ũ prescribes the displacement for some part of the body s boundary. In contrast, external stresses or traction forces are imprinted via a Neumann condition. For a total Lagrangian description, i.e., referring to the reference configuration, it reads F SÑ = t 0, with the unit normal vector Ñ. All in all, the strong form of the structural problem for the undeformed body B 0, with a boundary B 0 = B0 D B0 N, and a time span (0, T ) reads[9] ρ u tt = Div(F S) + ρ b 0 in B 0 (0, T ), (4a) E = 1 ( F T F I ) 2 in B 0 (0, T ), (4b) S = C : E in B 0 (0, T ), (4c) u = ũ on B D 0 (0, T ), (4d) F SÑ = t 0 on B N 0 (0, T ), (4e) u(x, t = 0) = u 0 (X) in B 0. (4f) Note that the equation of motion (4a) has been reformulated with respect to the undeformed configuration and the initial condition (4f) has been added, where u 0 (X) 0 is the most common case by far. In order to solve the structural problem (4), it is transformed into the weak form, in literature often referred to as the principle of virtual work[1, 9]: Find u(x, t) S t = {u H 1 (0, T ) u = ũ on B0 D t} such that Π(u, δu) = S : δe dv + ρ(u tt b 0 ) δu dv t 0 δu da = 0 δu S t 0. (5) B 0 B 0 B N 0

10 Seminar Thesis 4 It states a balance of the inner and outer stresses resulting from any virtual displacement δu S t 0 = {δu H1 (0, T ) δu = 0 on B D 0 t}, with the Sobolev space H 1 [9]. Although this postulation refers to the deformed configuration, the total Lagrangian formulation allows us to perform the integration over the undeformed body B 0. The discretization of the weak formulation, performed by means of isogeometric analysis, see Sec.2.1.2, results in a system of time-dependent nonlinear ordinary differential equations (ODEs), that reduces to a set of algebraic equations in case of steady simulations. Assuming the solution, i.e., the vector of discrete displacements U, is known for a time level t and to be determined for t + t, the system has the form[1] M t+ t Ü + t+ t K(U) t+ t U = t+ t F, (6) with the mass matrix M, the stiffness matrix K(U), and the right-hand side F. The upper left indices represent the time level the quantities are referring to. Due to the dependence of the stiffness matrix on the displacement and the concomitant nonlinearity the Newton-Raphson scheme is employed as an iterative procedure to solve the matrix form (6) [1]: M t+ t Ü (i) + t+ t K (i 1) U (i) = t+ t F t+ t R (i 1), t+ t U (i) = t+ t U (i 1) + U (i). Therein, the upper right indices denote the iteration level and, in particular, R (i 1) represents the inner stresses determined in the previous iteration step. The iteration is initialized with the (converged) solution of the previous time step, i.e., t+ t U (0) = t U, t+ t K (0) = t K, t+ t R (0) = t R. Note that in the steady case all time-dependent terms drop and only one pseudo time step is computed. For unsteady simulations, however, the remaining time integration is performed by means of the Bossak-α scheme[6, 17]. (7a) (7b) Isogeometric Analysis The spatial discretization of the shell structure is performed based on the so-called isogeometric analysis (IGA) introduced by Hughes et al.[7, 16]. The primary goals of this concept are to ensure geometrical exactness and entrench a stronger linkage between computer-aided design (CAD) and computer-aided engineering (CAE), i.e., the numerical analysis. While it in principle constitutes a variant of the finite element method, the fundamental difference lies in the main idea of isogeometric analysis: Instead of approximating the known geometry, e.g., by polynomial basis functions, IGA employs a basis capable of exactly representing the geometry[7]. Hence, the same basis the geometry is represented by in the CAD model is chosen. Due to their great flexibility regarding the shape of the geometry as well as their numerical stability and efficiency, Non-Uniform Rational B-Splines (NURBS) have become the industrial standard for the representation, design, and data exchange of geometrical information in modern CAD[21]. That is why the major part of the research work in isogeometric analysis so far has been focused on NURBS-based approaches.

11 Seminar Thesis 5 Non-Uniform Rational B-Splines (NURBS) Mathematically, a NURBS is a linear combination of its n control points B I and basis functions N I,p (ξ) of the NURBS degree p, that are defined in the parameter space ξ. For example, a NURBS curve or surface is expressed as[7] C(ξ 1 ) = n N p I (ξ1 )B I, S(ξ 1, ξ 2 ) = I=1 n I=1 N p,q I (ξ 1, ξ 2 )B I, (8) respectively, where the higher-dimensional basis functions N p,q I (ξ 1, ξ 2 ) are constructed as products of the one-dimensional functions. As NURBS are a generalization of B-Splines, their basis functions are derived, by introducing the weights w I, via[21] N p I (ξ) = w I Ñ I,p (ξ) n K=1 w KÑK,p(ξ) (9) from the B-Spline basis functions Ñi,p(ξ), which in turn are piecewise polynomials defined by the Cox-de Boor formula[21]: 1 if ξ i ξ < ξ i+1, Ñ i,0 (ξ) = (10a) 0 otherwise, Ñ i,p (ξ) = ξ ξ i Ñ i,p 1 (ξ) + ξ i+p+1 ξ Ñ i+1,p 1 (ξ). ξ i+p ξ i ξ i+p+1 ξ i+1 (10b) Their behavior strongly depends on the knot vector Ξ = (ξ 0 = a, ξ 1,..., ξ n+p+1 = b), a vector of non-decreasing parametric coordinates ξ i distributing the parameter space, ξ [a, b], into distinct knot spans, which form the elements in an IGA context. Due to their construction NURBS basis functions inherit many properties from their B-Spline counterparts, including those making them, beyond their geometrical properties, suitable for numerical analysis, too, as in particular their local support and partition of unity. For an in-depth discussion of NURBS Piegl et al.[21] is recommendable Isogeometric Reissner-Mindlin Shell Theory What are Shell Structures? Shell structures are thin-walled structures that are capable of carrying huge transversal stresses, while being rather sensitive to loads applied in normal, i.e., thickness, direction. The idea of shell structures is inspired by nature, where it can be found in both macroscopic and microscopic scales, e.g., egg shells or the membrane of a human body cell[14]. Due to their ability of providing light-weight, cost-efficient, but yet stable constructions, shell structures are among the most common structural elements in modern civil and aeronautical engineering[22].

12 Seminar Thesis 6 In order to perform numerical simulations of shell structures an associated mathematical model, a shell element, is required. The basic concept of shell elements is to make use of the small thickness by representing the structure as the midsurface plus some interpolation terms in thickness direction, lowering computational costs significantly in comparison to volume-based approaches. Shell Kinematics The shell element implemented in this thesis follows the Reissner-Mindlin shell kinematics, which employs a linear interpolation in thickness direction[5]. As a consequence, the shear strains only change linearly through the shell thickness[20]. In other words: Straight cross sections normal to the shell midsurface in the reference configuration stay straight under deformation, but in general not orthogonal, i.e., they are allowed to rotate around the midsurface. Therefore, in addition to the three global translational degrees of freedom Reissner-Mindlin shell elements feature two local rotational degrees of freedom around the tangent vectors of the midsurface[5]. A rotation around the shell normal, called drilling, is omitted and does not contribute to the shell kinematics. Beyond that, as a consequence of the thickness being very small, the inextensibility constraint demands the transverse normal strains to be zero, E 33 = 0 [9]. In this thesis, an isogeometric shell element is considered. Therefore, the midsurface is represented by a NURBS surface, i.e.,[9] X(ξ α ) = n N I (ξ α )X I, x(ξ α ) = I=1 n N I (ξ α )x I, (11) I=1 in the reference and the current configuration, respectively. Note that here, as well as in the remainder of the thesis, Greek letters represent the indices 1, 2, while Latin letters represent the range 1, 2, 3 whenever index notation is used. In combination with the linear interpolation over the shell thickness h by means of the parametric coordinate ξ 3 [ h/2, +h/2] the position of any material point in the shell structure is given by[9] X(ξ i ) = X(ξ α ) + ξ 3 D(ξ α ), x(ξ i ) = x(ξ α ) + ξ 3 d(ξ α ), (12) where the director vector D, or d respectively, has been introduced: In the undeformed configuration it corresponds to the normal vector of the shell midsurface, i.e., D(ξ α ) = N(ξ α ). Under deformation, however, the director vector stays similar to the normal vector to some extent, but in general the two vectors do not match, d(ξ α ) n(ξ α ). This disparity is due to the occurrence of shear strains inside the shell structure. The inextensibility condition E 33 = 0 is ensured by D(ξ α ) = d(ξ α ) = 1 [9]. As mentioned above, the shell element features two local rotational degrees of freedom around the tangents of the midsurface. Although a pair of tangent vectors is given by the NURBS derivatives G α (ξ α ) = X/ ξ α, it is convenient to introduce a local

13 Seminar Thesis 7 Cartesian coordinate system. The system s orthonormal basis is chosen to be as close as possible to the convected tangent vectors G α (and the normal N), i.e.,[9] 2 ) 2 ) A 1 = (Āξ 1 Ā ξ 2, A2 = (Āξ 1 + Ā ξ 2, A3 = N = D, (13) 2 2 with Ā ξ 1 = Ĝ1 + Ĝ2 Ĝ1 + Ĝ2, Ā ξ 2 = N Ā ξ 1 N Āξ 1 and Ĝ 1 = G 1 G 1, Ĝ 2 = G 2 G 2. The associated coordinates are denoted by θ i and can be used to express the position vector by analogy with Eqn.(12) as[9] X(θ i ) = X(θ α ) + θ 3 D(θ α ), x(θ i ) = x(θ α ) + θ 3 d(θ α ). (14) Moreover, the application of the chain rule yields the relation[9] ( ),α := ( ) θ α = J 1 ( ) αβ ξ, with the Jacobian J β αβ = θβ ξ = G α α A β, (15) between the derivatives with the respect to the local Cartesian coordinates θ α and the parametric coordinates ξ α. Both the director vectors and the local Cartesian coordinate system are evaluated as NURBS interpolations of control point associated vectors or systems, respectively[10]: n n n D(ξ α ) = N(ξ α )D I, d(ξ α ) = N(ξ α )d I, A i (ξ α ) = N(ξ α )A ii. (16) I=1 I=1 The nodal directors D I are determined via a closest point projection onto the midsurface. More evolved ansatzes are discussed in the works by Dornisch et al.[10, 12, 13]. I=1 Strains and Stresses An alternative formulation of the nonlinear kinematic relation (2), i.e., the Green-Lagrange strains, in terms of the local Cartesian coordinate system yields[9] E ij = 1 2 (x,i x,j X,i X,j ). (17) Plugging in the ansatzes for the position vectors from Eqn.(14) and reordering the terms with respect to the coordinate θ 3 leads to[9] E αβ = ε αβ + θ 3 κ αβ + (θ 3 ) 2 ϱ αβ, 2E α3 = γ α, E 33 = 0, (18) where the membrane strains ε αβ, the curvatures κ αβ, the second-order curvatures ϱ αβ, and the transverse shear strains γ α are defined by[9] ε αβ = 1 ) ( x,α x,β X,α X,β, 2 κ αβ = 1 ) ( x,α d,β + x,β d,α X,α D,β X,β D,α, 2 ϱ αβ = 1 2 (d,α d,β D,α D,β ), γ α = x,α d X,α D. (19a) (19b) (19c) (19d)

14 Seminar Thesis 8 Neglecting the second-order curvatures ϱ αβ, the Green-Lagrange strains of the shell midsurface are represented by the vector ε = (ε 11, ε 22, 2ε 12, κ 11, κ 22, 2κ 12, γ 1, γ 2 ) T, that can be used to reformulate the stress-strain integral as[9] S : E dv = S ij E ij µ G dθ 3 da = ε T σda. (20) B 0 θ 3 Ω 0 Ω 0 Therein, µ G denotes the determinant of the shifter tensor[9] that maps quantities from the shell midsurface Ω 0 to the shell body B 0 in the reference configuration. The vector σ = (n 11, n 22, n 12, m 11, m 22, m 12, q 1, q 2 ) T represents the 2nd Piola-Kirchhoff stresses and contains the membrane forces n αβ, the bending moments m αβ, as well as the shear forces q α, which are given by[9] n αβ = S αβ µ G dθ 3, m αβ = S αβ µ G θ 3 dθ 3, q α = S α3 µ G dθ 3. (21) θ 3 θ 3 θ 3 In Sec the linear stress-strain relation (3) has been introduced. Making use of linear elasticity theory for an isotropic material allows its reformulation with respect to the strain and stress vectors introduced above[9]: σ = Dε with D = θ 3 µ G C P θ 3 C P 0 θ 3 C P (θ 3 ) 2 C P C S where the matrices C P = E 1 ν 0 ( ) E ν 2 ν 1 0, C S = κ S 2(1 + ν) ν dθ 3, (22) model the translational and rotational kinematic part, respectively. Therein, E and ν denote the constant material parameters Young s modulus and Poisson ratio. The shear correction factor κ S is aimed at taking the difference between the assumed constant transverse shear stresses and their real counterparts into account. For rectangular cross sections its value is given as κ S = 5 [9]. 6 Now that the dependence on the local Cartesian coordinate in thickness direction θ 3 has been shifted from the stress and strain measures to the material matrix, a preintegration in thickness direction, i.e., over θ 3, is performed (from h to + h ). The 2 2 assumption of a constant shifter tensor with µ G = 1, which is convenient for thin shells[9], allows this integration to be conducted analytically, yielding[9] D = hc P 0 0 h 0 3 C 12 P hc S. (23) For thick shells the exact determinant of the shifter tensor should be considered, requiring a numerical evaluation of the integral in Eqn.(22). However, this option has not been implemented in the course of this seminar thesis.

15 Seminar Thesis 9 Geometries with Kinks While the Reissner-Mindlin approach of featuring two rotational degrees of freedom referring to a locally defined Cartesian coordinate system behaves well for smooth geometries, the occurrence of kinks corrupts the well-defined and unique assignment of the shell normal and hence the local Cartesian coordinate system. At this point, a major advantage of the NURBS parametrization is that it directly provides information about its continuity, allowing a classification of control points in terms of smoothness in a preprocessing step: If the associated parametric coordinate, e.g., the projection, corresponds to a knot of a multiplicity equal to the NURBS degree, m = p, a kink is possible from a parametric point of view, and hence the angle between the adjoining elements is compared to some small limit angle marking an upper bound for the classification as smooth [11]. For control points on a kink, three global rotational degrees of freedom are assigned as well as multiple directors and local Cartesian coordinate systems. The mapping from the local to the global Cartesian coordinates is therefore established via the transformation matrices[9] ) (a 1I a 2I for smooth control points, T 3I = ) (24) (a 1I a 2I a 3I for control points on a kink. This way, more tedious approaches like adding artificial stiffness are avoided[11]. Rotational Update Both the nodal director vectors and the nodal local Cartesian coordinate systems are updated from the reference to the deformed configuration by means of Rodrigues formula[9]: with the rotation tensor R I being defined by[9] d I = R I D I, a ii = R I A ii, (25) R I = R(ω I ) = 1 + sin ω I Ω I + 1 cos ω I Ω 2 ω I ωi 2 I, where ω I = ω I, (26a) 0 ω I3 ω I2 Ω I = Ω(ω I ) = skew(ω I ) = ω I3 0 ω I1. (26b) ω I2 ω I1 0 The update requires the axial vector of rotation ω I, which is determined from the rotational solution of the previous iteration steps. The implemented isogeometric shell element employs an multiplicative update technique, that is[9] R I (i) = R I R I (i 1) with R I = R( ω I ). (27)

16 Seminar Thesis 10 The increment of the axial vector of rotation ω I is the analogue of the increment of the rotational solution β I obtained in the previous iteration step for the considered control point, but in contrast it refers to the global Cartesian coordinate system. Thus, T 3I β I for smooth control points, ω I = (28) β I for control points on a kink, holds[11], as the rotational solution for control points on a kink already features three global rotational degrees of freedom. Note that here the transformation matrix T 3I is constructed based on the local coordinate system of the previous step as it has to be consistent to the rotational solution increment β I. Weak Form for the Isogeometric Shell Element If the relations of shell theory introduced so far are incorporated, the weak form of the structural problem, Eqn.(5), can be reformulated as[9]: Find v( X, t) such that δε T σ da + δv T ρh ( v b ) 0 da δv T t 0 ds = 0 δv. (29) Ω 0 Ω 0 Here, the vector v = (u ω) T combines the translational displacements u of the shell midsurface with the rotational degrees of freedom ω. Analogously, the vector of body forces b 0 as well as the traction forces t 0 are formulated with respect to the shell surface now, and therefore include rotational loads, i.e., moments, to ensure consistency. The Neumann boundary of the shell midsurface Ω 0 is denoted by Γ N 0. As a result of this pre-integration in thickness direction, the dimensionality of the domain has been reduced from the whole structural body to the shell midsurface only, drastically decreasing the computational costs for numerical quadrature. For this quadrature both the options of a full and a reduced Gauss integration are provided[9]. Γ N 0 Assembling the Element Matrices As mentioned in Sec.2.1.1, the spatial discretization yields a nonlinear matrix system of the form M Ü + K(U) U = F, see Eqn.(6). Analogously to the standard finite element procedure, the involved global matrices as well as the vector F are assembled element by element. For the implemented isogeometric shell element the element-level stiffness matrix K e = ( ) K e IK is computed via the numerical integration[9] I,K K e IK = Ω e ( BI T DB K + G IK ) da, (30)

17 Seminar Thesis 11 where the indices I and K run over all control points of the current element Ω e. The control point associated linear strain-displacement matrices B I have the form[9] N I,1 x Ț 1 0 N I,2 x Ț 2 0 N I,1 x Ț 2 + N I,2 x Ț 1 0 N I,1 d Ț 1 x Ț 1 B I = N I,1T I N I,2 d Ț 2 x Ț 2 N with T I = skew(d I ) T 3I. (31) I,2T I N I,1 x Ț 2 + N I,2d Ț 1 x Ț 1 N I,2T I + x Ț 2 N I,1T I N I,1 d T x Ț 1 T I N I,2 d T x Ț 2 T I Nonlinear stiffness terms are taken into account by the initial stress matrix G, that is constructed based on the node-level matrices[9] ( ) ˆn IK 1 ˆm uβ IK G IK = + ˆquβ IK ˆm βu IK + ˆqβu IK ˆmββ IK,1 (h1 ) + ˆm ββ IK,2 (h2 ) + ˆq ββ. (32) IK (hq ) The shorthand notations used in Eqn.(32) for the sake of observability are determined by the following set of relations: ˆn IK = n 11 N I,1 N K,1 + n 22 N I,2 N K,2 + n 12 (N I,1 N K,2 + N I,2 N K,1 ), ˆm uβ IK = [ m 11 N I,1 N K,1 + m 22 N I,2 N K,2 + m 12 (N I,1 N K,2 + N I,2 N K,1 ) ] T K, ˆq uβ IK = ( ) ( ) T ( T q 1 N I,1 + q 2 N I,2 TK, ˆm βu IK = ˆm uβ KI, ˆq βu IK = ˆq KI) uβ, h q = q 1 x,1 + q 2 x,2, h 1 = m 11 x,1 + m 12 x,2, h 2 = m 12 x,1 + m 22 x,2, M I (h) = 1 2 (d I h + h d I ) (d I h) 1, ˆm ββ IK,α (h) = δ IKN I,α T T 3I M I(h)T 3I, ˆq ββ IK (h) = δ IKN I T T 3I M I(h)T 3I, where denotes the outer product of two vectors. Similarly, the element-level mass matrix M e is assembled via the node-level matrices ( ) M e IK Ω = 1 0 ρn I N K h e h2 T T da, (34) I T K where 1 denotes the identity matrix of size 3. The vector F in the matrix form represents the external loads and is determined by[9] F e I = F e ext,i Ω = N I ρ h b 0 + N I t 0. (35) e In the course of the iterative solution procedure in Eqn.(7), the vector of the inner stresses computed in the previous step R, which is assembled via the nodal vectors[9] Γ e N R e I = F e int,i Ω = B T I σ da, (36) e is required, too, as it has to be subtracted from the external loads F to form the residual right-hand side in Eqn.(7a).

18 Seminar Thesis Computational Fluid Dynamics Navier-Stokes Equations Within the conducted simulations the physical behavior of fluids is modeled by the unsteady Navier-Stokes equations for an incompressible Newtonian fluid. The strong form of the associated initial-boundary value problem for a spatial domain Ω, bounded by the Lipschitz-continuous boundary Γ = Ω, and a time frame (0, T ) reads[8] v t ν 2 v + (v ) v + p = b in Ω (0, T ), (37a) v = 0 in Ω (0, T ), (37b) v = v D on Γ D (0, T ), (37c) pn + ν (n ) v = t on Γ N (0, T ), (37d) v(x, 0) = v 0 (x) in Ω, (37e) where ν denotes the kinematic viscosity and b the resultant force of all body forces acting on the fluid. The unknowns are the velocity field v and the kinematic pressure p = p/ρ, i.e., pressure divided by density. While Γ D is a Dirichlet boundary, a Neumann condition is applied to the remainder Γ N of the boundary, in which n is the outer normal vector and t the prescribed traction. The initial velocity field is given by v 0 (x) Solution Strategies The applied solver XNS is a finite element flow solver developed at CATS RWTH. Its main field of application are problems involving moving boundaries, e.g., free-surface flows or fluid-structure interaction. As the scope of this seminar thesis did not include any adjustments or extensions to the procedure of determining the flow solution, a detailed discourse on the performed steps is excluded here for the sake of conciseness, making way for a short outlook on the major solution strategies employed instead: The unsteady Navier-Stokes problem in Eqn.(37) is discretized by a stabilized space-time formulation, i.e., finite elements are applied in both space and time. In order to handle domains altering over time, the mesh is adapted to moving boundaries by means of interface-tracking. The resulting difference in the element sizes is compensated by the elastic mesh update (EMUM). For further information and theoretical background on the employed solution procedures the works by Behr[2], Behr et al.[3], as well as Elgeti et al.[15] are recommended.

19 Seminar Thesis Fluid-Structure Interaction Coupling Conditions When simulating the interaction between a fluid and a structure via a coupling interface Γ F S, i.e., the wetted body surface, the need for boundary conditions linking the two solution fields in a physically reasonable way, called coupling conditions, arises: The kinematic coupling condition[4] u F (t, x) = u S (t, x) t 0, x Γ F S (38) requires the displacements u F and u S of the coupling interface, to match for the flow field and the structure for all times t. The equivalence of the corresponding interface velocities, i.e., u F = u S, is a direct consequence. The dynamic coupling condition[4] σ F (t, x) n = σ S (t, x) n t 0, x Γ F S (39) ensures the conservation of stresses exchanged via Γ F S in normal direction n. The normal component of the stresses σ F the fluid imprints on the coupling interface due to static pressure as well as dynamic terms has to be equivalent to the normal component of the surface loads σ S the rigid body is experiencing. If these two coupling conditions are satisfied, the conservation of the mechanic power exchanged between the fluid and the structure directly follows[4] Partitioned Solution Algorithm At CATS RWTH a partitioned algorithm is used to run numerical simulations of fluidstructure interaction, i.e., two distinct solvers are employed for the fluid (e.g. XNS) and the structure (e.g. FEAFA) and connected via the Aeroelastic Coupling Module (ACM) that handles the data exchange between the two fields. The main advantage of this concept is its great modularity and the concomitant flexibility in terms of the applied solvers, which, however, comes at the price of additional effort that has to be made in order to ensure the coupling conditions to hold[4]. This task can be subdivided into two categories[23].: The spatial coupling primary is responsible for a reasonable projection of data between the non-conforming meshes of the fluid and the structure, while the goal of temporal coupling is to ensure consistency of the two solution fields before going on to the next time step, e.g., by running an iterative coupling procedure until convergence is reached. More information on partitioned algorithms for FSI problems can be found for example in the works of Braun[4], Reimer et al.[23], as well as Matthies et al.[19].

20 Seminar Thesis 14 3 Results 3.1 Structural Mechanics As the first and major step in validating the functionality of the implemented nonlinear isogeometric shell element it is employed to purely structural benchmark test cases in the following Cantilever Beam Subjected to End Moment Setup The first test case considered is a cantilever beam that is clamped at one end, while a uniform line moment is subjected to the opposing end. The general setup of this steady test case is illustrated in Fig.(1). The beam has a length of L = 12, a width of B = 1, and Figure 1: The setup and parameters of the first structural benchmark test. a thickness of t = 0.1. The corresponding material is defined by a Young s modulus of E = and a Poisson ratio of ν = 0.0. Due to the latter choice and the simple setting of the test case, an analytical solution is provided by beam theory: The displacements u and w of the loaded end in x- and z-direction, respectively, are given as functions of the applied moment M by[24] u(m) L = M ( 0 M M sin M 0 ) 1 and w(m) L = M 0 M ( 1 cos ( )) M, (40) with M 0 = 25/3, yielding a fully rolled up beam for a moment of M max = 2πM 0. This moment M max is applied gradually in five equal load steps. For each of these steps the numerical results obtained will be compared to the analytical reference solution. M 0

21 Seminar Thesis 15 Discretization As this test case is aimed at validating both the employed structural model and its implementation rather than studying the convergence behavior for varying discretizations, the discussion of the numerical results is restricted to those obtained for the rather fine (in length direction) structural mesh of nonlinear isogeometric shell elements of NURBS degree 2. Note, however, that coarser discretizations are capable of yielding a satisfying accuracy for the rolled up cantilever beam as well. Results In Fig.(2) the numerical results obtained for both the displacements in x- and z-direction of the loaded end are compared to their analytical counterparts, represented by the dashed lines. For the sake of observability the absolute values are plotted. Figure 2: The numerical results obtained for the five load steps compared to the analytical ones. One can state that the results are in good compliance to the reference solution. In particular, the displacements computed for the full moment M max, i.e., u L and w 0, indicate that the beam is fully rolled up as predicted by the analytical solution. This observation is confirmed by Fig.(3), which shows the deformed beam for the five load steps. Taken as a whole, the test case of the cantilever beam subjected to an end moment indicates a correct modeling and implementation of the transfer of stresses and, in particular, bending moments through the shell structure.

22 Seminar Thesis 16 Figure 3: The structural deformation computed for the five load steps Channel Section Beam Subjected to Point Load Setup The next structural test case is more complex in the sense that it involves the handling of geometries with kinks: A clamped channel section beam, with the profile shape shown in Fig.(4b), is loaded by a constant point load at its free end. Again, the test case is steady and no gravity is acting on the structure. Both the test case setup and its parameters are depicted in Fig.(4). The beam has a length of L = 900, a Young s modulus of E = , and a Poisson ratio of ν = 0.3, while its cross section exhibits a height of H = 28.4, a width of B = 9.5, and a thickness of t = 1.6 or s = 1.0, respectively. In order to avoid numerical instabilities the point load of F = is imprinted step by step in 30 even load steps. As no analytical solution is available for the test case of the channel section beam, the numerical results obtained will be compared to those from literature, in particular the dissertation of W. Dornisch[9].

23 Seminar Thesis 17 (a) The measurements and parameters. (b) The beam s profile. Figure 4: The test case setup for the channel section beam. Discretization Alongside with the general validation of the shell s handling of kinks, the test case is focused on the convergence behavior in dependence on the numerical discretization applied. The coarsest mesh for the performed computations consists of 18 nonlinear isogeometric shell elements in length direction and 5 in lateral direction, of which 3 elements are employed to model the main section of the beam, whereas one element is used per wing. Finer meshes are derived by uniformly increasing the number of elements in each direction and for each section, including the wings, by the same factor. For example, the next finer mesh would feature elements, and so on. The convergence is investigated for NURBS degrees of 2 and 4. Results To give an impression on the resulting structural deformation, Fig.(5) visualizes the deformed channel section beam, computed with shell elements of degree 2. Clearly, the picture confirms that the test case involves large deformations and rotations, making a geometrically nonlinear structural model inevitable. Beyond that, only a qualitative assessment of the deformation, if any, is possible from Fig.(5). As to this, one might state that the deformation looks reasonable from a physical point of view. For a more profound discussion of the numerical results a closer look on the displacement of the loading point in z-direction in dependence on the number of elements employed is taken in Fig.(6). Here, the solution for nonlinear isogeometric shell elements of NURBS degree 4 provided by Dornisch[9] serves as a reference and is marked by the dashed line. To simplify an assessment of the results a relative error of η = 0.01 is illustrated as well. The plot indicates that the results for both a NURBS degree of 2 and 4 converge to the reference solution. In particular for a degree of 4, however, the implemented shell element so far slightly underestimates the structural

24 Seminar Thesis 18 Figure 5: The structural response of the channel section beam. Figure 6: The z-displacement of the loading point over the number of elements in length direction. deformation in comparison to the convergence study conducted by Dornisch[9]. Nevertheless, the test case of the channel section beam under point load adverts to a reasonable implementation and functionality of the new shell element.

25 Seminar Thesis Fluid-Structure Interaction Initially, the task specification of this seminar thesis included the application of the implemented nonlinear isogeometric shell element to numerical simulations of fluidstructure interaction. Unfortunately, unforeseen problems with the involved flow and structural solver as well as issues concerning stability and convergence of the partitioned algorithm coordinating their interplay raised the effort required to realize this goal to a level way beyond the scope of this thesis. As a consequence, the successful simulation of the FSI test case shortly outlined in the following had to be postponed to a later work or thesis Sloshing Tank In this test case a cylindrical tank is filled with a fluid up to a certain level. The tank wall is modeled by a structural mesh of nonlinear isogeometric shell elements. In order to simulate an earthquake, the bottom of the tank starts a prescribed periodic horizontal movement. Both the resulting unsteady structural deformation and inertia terms cause a sloshing movement of the fluid. Vice versa, the hydrostatic pressure as well as dynamic CFD loads imprint additional forces on the tank wall, establishing the interdependency typical for applications of fluid-structure interaction. Note that this test case is not a benchmark test, but rather designed for a qualitative check of the mechanisms involved in the partitioned algorithm for a three-dimensional FSI simulation, e.g., the spatial and temporal coupling of the flow solver XNS with the structural solver FEAFA via the coupling module ACM. Figure 7: The FSI test case of a sloshing tank.

26 Seminar Thesis 20 4 Conclusion The main goal of this seminar thesis has been the implementation of a nonlinear isogeometric shell element capable of predicting the structural response of thin-walled structures. The employed structural model is geometrically nonlinear and hence designed to adequately handle large deformations and rotations, but only small strains as the stress-strain relation is still assumed to be linear. Although several adjustments have been made the shell element implemented in the course of this thesis is following the element presented by W. Dornisch in his PhD thesis[9] to a great extent. The biggest problems encountered during the implementation were related to the treatment of the rotational degrees of freedom, in particular the missing drilling rotation as well as the handling of geometries with kinks and the related varying number and meaning of rotational degrees of freedom. Concerning this, methods had to be found that yield stable and reliable results, while avoiding locking effects or artificial stiffnesses. After the implementation of the shell element its functionality has been validated based on structural benchmark test cases. The ones discussed in this thesis are the cantilever beam subjected to an end moment presented in Sec and the channel section beam under point load covered in Sec Further simulations, including unsteady computations, have been run but are not covered in this text for the sake of brevity. The conclusion drawn from these structural test cases is that the adaptation of a nonlinear isogeometric shell element to the structural solver FEAFA has been successful, yielding physically reasonable deformations that bear comparison with results found in literature. Beyond the field of structural mechanics, however, originally the application of the new shell element to simulations of fluid-structure interaction was intended, e.g., the test case of the sloshing tank presented in Sec Unfortunately, the problems occurring in the course of the FSI part of the thesis, among others, in terms of stability and convergence, caused an extensive increase of the effort required to reach that goal, until it clearly surpassed the scope of this seminar thesis. As a consequence, the utilization of the implemented shell element in numerical simulations involving fluid-structure interaction had to be postponed to a future work. Nevertheless, one can conclude that the key task of the thesis has successfully been completed, that is the implementation of a geometrically nonlinear isogeometric shell element into the structural solver FEAFA.

27 Seminar Thesis 21 References [1] K.-J. Bathe. Finite Element Procedures. Prentice Hall, [2] M. Behr. Stabilized Finite Element Methods for Incompressible Flows with Emphasis on Moving Boundaries and Interfaces. PhD thesis, University of Minnesota, [3] M. Behr and F. Abraham. Free-Surface Flow Simulation in the Presence of Inclined Walls. Computer Methods in Applied Mechanics and Engineering, [4] C. Braun. Ein modulares Verfahren für die numerische aeroelastische Analyse von Luftfahrzeugen. PhD thesis, RWTH Aachen University, [5] D. Chapelle and K.-J. Bathe. The Finite Element Analysis of Shells - Fundamentals. Springer-Verlag Heidelberg Dordrecht London New York, 2nd edition, [6] J. Chung and G. M. Hulbert. A Time Integration Algorithm for Structural Dynamics with Improved Numerical Dissipation: The Generalized-α Method. Journal of Applied Mechanics, [7] J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs. Isogeometric Analysis - Toward Integration of CAD and FEA. WILEY, [8] J. Donea and A. Huerta. Finite Element Methods for Flow Problems. WILEY, [9] W. Dornisch. Interpolation of Rotations and Coupling of Patches in Isogeometric Reissner-Mindlin Shell Analysis. PhD thesis, RWTH Aachen University, [10] W. Dornisch and S. Klinkel. The Interpolation of the Director Vector for Isogeometric Reissner-Mindlin Shell Analysis. Conference Paper, 6th European Congress on Computational Methods in Applied Sciences and Engineering, Vienna (Austria), [11] W. Dornisch and S. Klinkel. Treatment of Reissner-Mindlin Shells with Kinks without the Need for Drilling Rotation Stabilization in an Isogeometric Framework. Computer Methods in Applied Mechanics and Engineering, [12] W. Dornisch and S. Klinkel. On the Choice of the Director for Isogeometric Reissner-Mindlin Shell Analysis. Conference Paper, 10th World Congress on Computational Mechanics, São Paulo (Brazil), [13] W. Dornisch, S. Klinkel, and B. Simeon. Isogeometric Reissner-Mindlin Shell Analysis with Exactly Calculated Director Vectors. Computer Methods in Applied Mechanics and Engineering, [14] R. Echter. Isogeometric Analysis of Shells. PhD thesis, University of Stuttgart, 2013.

28 Seminar Thesis 22 [15] S. Elgeti and H. Sauerland. Deforming Fluid Domains within the Finite Element Method. Computational Methods in Engineering, [16] T. J. R. Hughes, J. A. Cottrell, and Y. Bazilevs. Isogeometric Analysis: CAD, Finite Elements, NURBS, Exact Geometry and Mesh Refinement. Computer Methods in Applied Mechanics and Engineering, [17] D. Kuhl and A. Crisfield. Energy-Conserving and Decaying Algorithms in Non- Linear Structural Dynamics. International Journal for Numerical Methods in Engineering, [18] B. Markert. Porous Media Viscoelasticity with Application to Polymeric Foams. PhD thesis, University of Stuttgart, [19] H. G. Matthies and J. Steindorf. Partitioned Strong Coupling Algorithms for Fluid-Structure Interaction. Computers & Structures, [20] A. H. Nayfeh and P. F. Pai. Linear and Nonlinear Structural Mechanics. WILEY-VCH, [21] L. Piegl and W. Tiller. The NURBS Book. Springer-Verlag Berlin Heidelberg New York, 2nd edition, [22] E. Ramm and W. A. Wall. Shell Structures - A Sensitive Interrelation between Physics and Numerics. International Journal for Numerical Methods in Engineering, [23] L. Reimer, C. Braun, G. Wellmer, M. Behr, and J. Ballmann. Development of a Modular Method for Computational Aero-Structural Analysis of Aircraft. Management and Minimisation of Uncertainties and Erros in Numerical Aerodynamics, [24] K. Sze, X Liu, and S. Lo. Popular Benchmark Problems for Geometric Nonlinear Analysis of Shells. Finite Elements in Analysis and Design, 2004.