A Variationally Consistent Mesh Adaptation Method for Triangular Elements in Explicit Lagrangian Dynamics

Transcription

1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 28; : 39 [Version: 22/9/8 v2.2] A Variationally Consistent Mesh Adatation Method for Triangular Elements in Exlicit Lagrangian Dynamics Sudee K. Lahiri, Javier Bonet 2, and Jaime Peraire Aerosace Comutational Design Laboratory, Deartment of Aeronautics & Astronautics, MIT, Cambridge, USA. 2 Civil & Comutational Engineering Center, School of Engineering, Swansea University, UK. SUMMARY In this aer a variational formulation for mesh adatation rocedures, involving local mesh changes for triangular meshes, is resented. Such local adative changes are very well suited for exlicit methods as they do not involve significant comutational exense. They also greatly simlify the rojection of field variables from the old to the new meshes. Crucially, the variational nature of the formulation used to derive the equilibrium equations at stes where adatation takes lace ensures that conservation of linear and angular momentum is obtained []. Several examles in 2-D showing the alication of the roosed adative algorithms are used to demonstrate the validity of the methodology roosed. Coyright c 28 John Wiley & Sons, Ltd. key words: Variationally consistent, mesh adatation, exlicit, sace-time discretization, triangular elements. Introduction Raid dynamics encomasses a significant section of continuum mechanics roblems. Several industrial henomena involve raid dynamics of solids, for examle forging, machining, crashtests, collision modeling and many others. Comutational simulations of such roblems are used in various engineering analysis and design. These roblems involve large deformations and rotations along with comlex material behavior. Hence these roblems are inherently non-linear. Due to high velocities (of the order of seed of sound in the material), large meshes and many small time-stes are used for satial and temoral accuracy. Hence exlicit timeintegrators become advantageous in such alications. Several codes have been develoed and used for such roblems [2, 3, 4, 5], based on exlicit methods. The main challenges in these Corresondence to: Javier Bonet, Civil & Comutational Engineering Center, School of Engineering, Swansea University SA2 8PP, UK.(j.bonet@swansea.ac.uk) Contract/grant sonsor: Sandia National Laboratory, USA.; contract/grant number: Doc. No. 52 under A26 Coyright c 28 John Wiley & Sons, Ltd.

2 2 S. K. LAHIRI ET AL. numerical roblems lie in the roer modeling of large deformations and rotations, contact, and comlex non-linear material behavior. Mesh distortions, encountered due to large deformations lead to lack of accuracy of the solution. Mesh adative time integration can be used to reduce mesh distortions and increase the accuracy of the solution. Such use of mesh adatation has been limited, since these udates add errors to the solution. Existing mesh-adative methods combined with common time integration techniques do not ensure conservation of momentum which may lead to errors over many time-integration stes. Hence, it is desired that such mesh adatation methods conserve global momentum which would allow use of adatation in reducing mesh distortions and also increase the accuracy of the solution. An imortant asect of a time-integration method in dynamics alications is its ability to conserve mass, momentum (linear and angular) and energy, which leads to more hysically consistent solutions. Methods which do not have good conservation roerties, develo large errors over many time integration stes. Tyically, dynamics in solids are modeled from a Lagrangian formulation of the equations of motion. Hence mass conservation is automatically satisfied in such methods. Exact conservation of global energy is hard to obtain using exlicit integrators. But global momentum (linear and angular) conservation is ossible. The exlicit time-integrator, the Central Difference Scheme (also called the Lea-Frog Method), is found to conserve global momentum exactly. Existing codes [4, 5] have emloyed this method with great success. Recent research [, 6] has shown that time-integration methods develoed from a variational rincile as that of Hamilton s rincile of stationary action, necessarily conserve linear and angular momentum. Such methods are commonly called as Variational Integrators or Variational methods. In this aer, toological changes for mesh adatation are develoed from Hamilton s rincile and sace-time discretization, leading to Variational Mesh Adatation which conserves the total momentum (linear and angular) of the discrete system... Literature review... Variational Framework Variational integrators have been develoed by several researchers [, 7, 8, 9,,, 2, 3], on the basis of Hamilton s rincile of stationary action, rather than discretizing the differential equations of motion in time. Hamilton s rincile dictates that the ath followed by a body reresents a stationary oint of the action integral of the Lagrangian over a given time interval [4, 5]. Variational integrators take advantage of this rincile by constructing a discrete aroximation of this integral which then becomes a function of a finite number of ositions of the body at each time ste. The stationary condition of the resulting discrete functional with resect to each body configuration leads to time steing algorithms that retain many of the conservation roerties of the continuum roblem. In articular, the schemes develoed in this way satisfy exact conservation of linear and angular momentum []. In addition, these algorithms are found to have excellent energy conservation roerties even though the exact reasons for this are not fully understood [, 6, 7, 8]. This class of variational algorithms includes both imlicit and exlicit schemes, and in articular, it includes some well-known members of the Newmark family [9]. A recent develoment in the area of variational integrators, is the develoment of asynchronous variational integrators []. The discrete energy gets comuted as the variation of the Lagrangian with resect to the time-ste. By altering the time ste locally, it has been shown in [], that variational integrators could have both momentum and energy conserving roerties but at the cost of

3 VARIATIONAL MESH ADAPTATION 3 being asynchronous. This aer discusses only synchronous time-ste methods...2. Mesh Adatation Mesh adatation has been an active area of research in solid and fluid mechanics comutations. There are three tyes of mesh adatation viz. : () r-adatation, where the number of nodes and number of elements remain same while the node locations or connectivities are changed [2], (2) h-adatation, where the elements are refined and de-refined locally or globally [2], and (3) -adatation, where the order of the interolation olynomial within the element is changed to resolve the solution locally [22]. The effectiveness of mesh adatation deends on the mesh-adative-mechanism, and the adatation criteria. Mesh-adative mechanisms might include local mesh changes or global remeshing. Global mesh changes, tyically involve, comlete remeshing and transfer of variables from the old mesh to the new mesh [23, 24]. Local mesh changes could be achieved using exlicit udates [25, 26]. Mesh changes involve node movement, changes in mesh connectivity, and coarsening and refinement of meshes. A detailed overview of such changes in meshes can be found in [27, 28]. Various such mesh udate methods exist, which are used by several researchers [4, 29, 3] with success. 2D remeshing based on the advancing front methods have been used in [3] for modeling ballistic enetration roblems. Severe mesh distortions encountered in 2D machining roblems have been handled in [3], based on comlete remeshing techniques. 2D mesh adatation for shear bands in lane strain can be found in [32, 33] Local coarsening and refinement based on mesh size has been discussed in [29] in alication to shear bands. 3D Mesh oerations are discussed in [29, 27]. Mesh adatations for metal forming can be found in [34, 35, 36]. 2D Imact roblems have been modeled using global remeshing and gradient based indictors in [37]. Mesh adatation has also been used in shae otimization of structures [38, 39]. The adatation criteria is chosen by the analyst. Tyically meshes are adated based on either some error-estimate or mesh skewness or some outut of interest. Various researchers [4, 4, 42, 43, 44, 23, 45] have described different error estimation techniques in their works. A commonly used error estimate by Zienkiewicz and Zhu, [46, 47], (Z 2 error estimate), uses the stresses within the element and describes a recovery rocess to obtain a reference stress. More recent develoments, [48, 49, 5] have used a new aroach for error-estimation based on the constitutive relation error. Error estimates based on variational constitutive udates can be found in [23]. Variational mesh adatation, where the error-estimate is obtained from a variational rincile is found in [23, 5, 52, 53]. Recently, some researchers [39, 54], have used the idea of configurational forces [55] for r-adatation, for alications in shae otimization. An overview of various error-estimation techniques and adatation criteria can be found in [45, 56]..2. Overview Section 2 of this aer reviews the variational framework from time ste integrators. The details of the sace-time discretization used later in our adative formulation are then resented. The derivation of the simle lea frog method using sace-time discretization is shown as an examle. In section 3, the sace-time discretization and the variational formulation are extended to incororate local mesh adatations. Local remeshing is achieved by four local oerations, viz.: () Diagonal Swaing, (2) Edge Slitting, (3) Node Movement and (4) Edge Collasing. Details of the above mechanisms are resented individually. Then,

4 4 S. K. LAHIRI ET AL. imlementation details of error-estimation, and adatation criteria are mentioned followed by examles demonstrating the erformance of the adatation methods. In section 6, A brief summary of the overall develoments of the research is resented, followed by suggestions of ossible future work. 2.. The Continuous Problem 2. Variational Formulation The motion under loading of a generic three dimensional body is considered. A reference configuration, Q R 3 is adoted, corresonding to the configuration of the body at time t =. The material coordinates X Q, are used to label the articles of the body. At any arbitrary time t, the osition of article X is given by the coordinate x, and in general, the motion of the body is described by a deformation maing, x = φ(x,t), () as illustrated in figure. In its reference configuration, the body has volume V and density ρ, whereas at a given time t, the body has volume V (t) and density ρ(t). Figure. Continuous systems 2.2. The Action Integral for non-dissiative systems For non-dissiative systems, both the internal and external forces in the system can be derived from a otential, and the motion between times t = and t, can be determine from Hamilton s rincile. To this end, a Lagrangian, L, is introduced, such that, L(x,ẋ) = K(ẋ) Π(x), where, K, denotes the kinetic energy, Π is the otential energy and ẋ = dx/dt is the material velocity. The otential energy can be generally decomosed into an internal elastic comonent, Π int, and a comonent accounting for the external conservative forces, Π ext. Thus, Π(x) = Π int (x) + Π ext (x).

5 VARIATIONAL MESH ADAPTATION 5 The action integral, S, is defined as the integral of the of the Lagrangian over the time interval considered, S = t L(x,ẋ)dt, (2) and Hamilton s rincile states that the deformation maing satisfying the equations of motion can be obtained by making the action integral stationary with resect to all ossible deformation maings which are comatible with the boundary conditions [5], where the Lagrangian L can be exressed in terms of the deformation and velocities in the following manner The Kinetic Energy, (K) The kinetic energy of the body is a function of the material velocity and can be written as: K(ẋ) = V 2 ρ ẋ 2 dv. (3) The Internal Potential Energy (Π int ) The internal otential energy deends on the constitutive relations of the materials in the system. In this research hyerelastic comressible Neo-Hookean materials are considered, which undergo large deformations and dislacements. Let F be the deformation gradient tensor which can be written as, F ij = x i X j i, j =,.., 3 The relevant kinematic quantities associated with the deformation gradient are the right Cauchy-Green tensor, C, the Jacobian, J, and the isochoric comonent of C, Ĉ, which are given by, C = F T F; J = det(f); Ĉ = J 2 3 C. For isotroic comressible Neo-Hookean materials, the internal otential energy can be exressed in terms of the Lame constant µ, and the bulk modulus κ as Π int (x) = π(f)dv V [ µ (tr(ĉ) ) = ] 2 κ(j )2 dv. (4) V The above exression is well suited for comressible or nearly incomressible materials, [57, 58] The External Potential Energy (Π ext ) The external otential energy includes the work done by the external body and surface forces. Π ext (x) = f b x dv f s x ds (5) V V Here, f b are the body forces (er unit volume), f s are the surface forces (er unit surface), and V denotes the section of the boundary, in the reference configuration, where the surface forces are alied.

6 6 S. K. LAHIRI ET AL Discretization in time Consider now a sequence of timestes t n+ = t n + t, n =,,..., N, where for simlicity a constant ste size has been taken. The osition of the body at each ste is defined by a maing x n = φ(x, t n ). A variational algorithm is defined by a discrete sum integral, S(x, x,..., x N ) N n= L n,n+ (x n, x n+ ) (6) where the discrete Lagrangian Integral L aroximates the integral of the continuum Lagrangian L over a timeste [], that is, L n,n+ (x n, x n+ ) tn+ t n tn+ t n L(x, ẋ)dt tn+ K(ẋ)dt Π(x)dt (7) t n Here, for simlicity, the case in which the Lagrangian is a function of x and ẋ only, is considered. Other cases, like the ones where the Lagrangian is deendant on ressure in addition to the osition and velocity are discussed in [6, 59]. The discrete Lagrangian Integral can be further slit into the Kinetic Energy Integral and the Potential Energy Integrals as: L n,n+ (x n, x n+ ) = K n,n+ (x n, x n+ ) tn+ t n Π(x)dt (8) where K n,n+ is an aroximation to the Kinetic Energy Integral t n+ t n K(ẋ)dt. There are many ways in which the aroximation (8) can be chosen, and, each one will lead to a different time integration algorithm. It has been shown in [6] that the aroximation for the Potential Energy Integral : tn+ Π(x)dt t Π(x n ) (9) t n where t = t n+ t n, leads to exlicit time marching algorithms with aroriate choice of the discrete Kinetic Energy Integral. Hence with exlicit methods in consideration, the discrete Lagrangian within two stes can be rewritten as: L n,n+ (x n, x n+ ) = K n,n+ (x n, x n+ ) t Π(x n ) () The stationary conditions of the discrete sum integral S with resect to a variation δv n of the body osition at time ste n are now given by, D n S[δv n ] = D 2 L n,n (x n, x n )[δv n ] + D L n,n+ (x n, x n+ )[δv n ] = δv n, () where D i denotes directional derivative with resect to i-th variable. The above equation reresents the statement of equilibrium at ste n and will enable the ositions at ste n + to be evaluated in terms of ositions at n and n. Rewriting the stationary conditions in terms of the Kinetic and Potential Energy Integrals we obtain: D 2 K n,n (x n, x n )[δv n ] + D K n,n+ (x n, x n+ )[δv n ] t D Π(x n )[δv n ] = δv n (2)

7 VARIATIONAL MESH ADAPTATION Discretization in time and sace So far in this aer, discretizations in time have been discussed. In this section a simle satial discretization is introduced using 3 noded triangular elements. Based on triangular elements, the osition vector x n e in an element e, can be written as: x e n = N e ax a n (3) where N e a are linear shae functions within an element e and x a n are the nodal osition vectors. The action integral as discretized in time in equation 6 now can be rewritten as: S = S(x a n; a =,...,N d ; n =,...,N) N L n,n+ (x a n, xa n+ ; a =,...,Nd ) (4) n= where N d are the number of nodes and N are the number of time stes. The Lagrangian within the time stes n and n + can be written as: L n,n+ (x a n, xa n+ ) = The stationarity condition then becomes: S x a n K n,n+(x a n, xa n+ ) t ( Π ext n (xa n ) + Πint n (xa n )) (5) = L n,n+ x a n + L n,n x a n = (6) which leads to the relations between the derivatives of the Kinetic and Potential Energy integrals as: K n,n+ x a n t Πext n x a n t Πint n x a n Now, each of the derivatives will be calculated searately. + K n,n x a n = (7) 2.5. The Potential Energy Integral First, the internal Potential Energy and its derivative with resect to x a n are calculated. The Potential Energy is a function of x a at time level n only due to the aroximation in Eqn. 9. Hence, for convenience, the time index n is droed for the rest of this section on Potential Energy Integral. Therefore, x a imlies x a n unless mentioned otherwise. In addition, the following index notation is used. Indices e and f are used to denote elements, indices a and b are used to denote nodes, and, indices i, j, k and l are used to denote vector directions in the current (satial) configuration and I, J, K, L are used to denote the directions of vectors in the reference (material) configuration. Reeated indices imly summation. First, the deformation gradient within the element e, is considered: F e = x X = xe a Ne a X where X is the osition vector of the reference (material) configuration. Note that since the shae functions are linear in the element the gradients are constant within an element hence (8)

8 8 S. K. LAHIRI ET AL. the deformation gradient is a constant within the element. Based on the Neo-Hookean model, the internal otential energy (Π int ), can be written as: Π int (x) = π(f e )dve (9) where e π(f e ) = µ 2 V e { tr( C ˆ } e ) 3 + κ 2 (J e ) 2 (2) J e = det(f e ); C e = F et F e ; b e = F e F et ; Ĉ e = J 2 3 e C e ; Therefore, the derivative of otential energy wrt. x can be written as : π(f) x a i = π(f) F : F x a i F = P : where P is the first Piola Kirchhoff stress tensor. The first Piola Kirchhoff stresses are related to the Cauchy stress tensor (also called the true stresses) by: Further simlifying using indicial notation, leads to: π x a i x a i (2) (22) σ = J P F T (23) = P il N e a x j F e jl (24) Now, introducing a global index of a node as, b, such that it is the a th node of element e, (reresented here as: (e, a) b ) and revisiting equation, 9 & 24, one can exress the derivative of the Potential Energy as: Π int (x) x b i = = (e,a) b (e,a) b V e V e π e (F e ) x a i dv e (25) P il N e a x j F e jl dv e (26) Now, changing the reference volume V to V a current volume one can obtain: Π int (x) Na e x b = P il FjL e Je dv e (27) i V e x j (e,a) b Substituting equation 23 into equation 27, one can obtain: Π int (x) Na e x b = σij e i (e,a) b V e x dv e j = Ti b = Tai e (28) (e,a) b

9 VARIATIONAL MESH ADAPTATION 9 where Ti b are the internal tractions at node b along direction i, and the Tai e are the elemental internal tractions at a th node of the element along direction i. Similar to the internal Potential Energy, it can be shown [59] that the external Potential Energy (5) would have similar derivatives: Π ext (x) x b i = = F b i (e,a) b V e = ρ N e a f b i dv e (e,a) b (e,a) b V e N e a f s i ds e F e ai (29) where fi s are external surface force er unit area, and fb i are the body forces er unit volume. Thus the final exression for the derivative of the Potential Energy with resect to the osition vector of a global node at time level n (x b n) is: Π(x n ) x b n = Πint (x n ) x b n + Πext (x n ) x b n = T b n F b n (3) 2.6. The Kinetic Energy Integral & Sace-Time Discretization In this section a sace-time discretization is adoted to formulate the Kinetic Energy Integral described in Eqn. 8. The sace-time integrations are conducted over the sace-time vector sace V R (undeformed reference configuration), and not on V R (deformed configuration). We begin with a single triangular element with unit thickness. c t n+ b 4 c a b t 3 Y 2 t n a X Figure 2. The sace-time-rism (left) and a generic sace-time-tetrahedron (right). Figure 2, shows the tyical sace-time volume of a single triangle. The triangle abc n and triangle abc n+ enclose a rismatic sace-time volume. This volume is further sub-divided into three tetrahedra. The task is to comute the kinetic energy integral K within each of the sacetime-tetrahedra, and then sum each of the contributions to comute the net integral within the sace-time-rism. To do so, a generic sace-time-tetrahedron (Fig. 2, (right)) is studied and the integral is comuted by first evaluating the constant velocity over the sace-time

10 S. K. LAHIRI ET AL. tetrahedron as: x = x(x,t) v n,n+ = dx dt (3) Where x is the osition vector, X is the reference osition vector and d dt is the total derivative. Note here, that for the Kinetic Energy Integral, total derivatives of osition vectors, are considered. In the general case, any quantity (scalar or vector) would have a similar treatment. First, a set of volume coordinates are introduced, analogous to the area coordinates in case of triangles. The volume coordinates, given by (ξ, ξ 2, ξ 3, ξ 4 ) attain values of at their corresonding nodes and zero at other nodes, ie., ξ i is one at node i and zero at all nodes j i. Any function linear in X, Y, t, say F(X, Y, t), can be interolated within the tetrahedron, based on its nodal values F a and shae functions N a = ξ a as F(X, Y, t) = F a ξ a. The coordinate transform between (X, Y, t) and (ξ, ξ 2, ξ 3, ξ 4 ) can be written as: X Y t Inverting this relation gives: ξ ξ 2 ξ 3 ξ 4 = = 6V X X 2 X 3 X 4 Y Y 2 Y 3 Y 4 t t 2 t 3 t 4 6V a b c 6V 2 a 2 b 2 c 2 6V 3 a 3 b 3 c 3 6V 4 a 4 b 4 c 4 ξ ξ 2 ξ 3 ξ 4 X Y t (32) (33) where a i s are the cofactor of the X i elements in the transformation matrix. Similarly b i s are the cofactors of the Y i elements, c i s are the cofactors of the t i elements, and V i s are one sixth the cofactor of each unit element in the transformation matrix. V is the volume of the tetrahedron given by: V = 6 X X 2 X 3 X 4 Y Y 2 Y 3 Y 4 t t 2 t 3 t 4 (34) Using chain rule, the derivatives of the function now can be written as : df dx i = F ξ j dξ j dx i (35) where X = [ X Y t] T. Thus, the time gradient of the function F can be written as: Now, since F is linearly interolated, df F = c j (36) dt 6V ξ j ( ) F ξ i = F i which leads to a simle relation for the time

11 VARIATIONAL MESH ADAPTATION derivatives: df dt = X X 2 X 3 X 4 Y Y 2 Y 3 Y 4 F F 2 F 3 F 4 X X 2 X 3 X 4 Y Y 2 Y 3 Y 4 t t 2 t 3 t 4 Similarly, assuming a linear interolation of x (= F(X, Y, t)) in sace and time, the velocity within the tetrahedron is obtained as a ratio of two determinants: v n,n+ = dx dt = X X 2 X 3 X 4 Y Y 2 Y 3 Y 4 x x 2 x 3 x 4 X X 2 X 3 X 4 Y Y 2 Y 3 Y 4 t t 2 t 3 t 4 Note here, that in the secial case where two nodes of a given sace-time-tetrahedron have the same reference coordinate (imlying the same oint) then, the velocity in the tetrahedron simly becomes (in this case assuming X = X 4 and Y = Y 4 ): (37) (38) v n,n+ = x 4 x t 4 t (39) This simlification leads to a criterion for the choice of subdivision of any generic sace-time volume. One should choose to sub-divide a given sace-time volume into as many tetrahedra with common nodes as ossible. This would lead to a simle velocity interolation within the tetrahedron. Now the Kinetic Energy Integral K is comuted within the sace-timetetrahedron: K tet n,n+ = V 2 ρ (v n,n+ v n,n+ )dv (4) In the case of a tetrahedron with common nodes this volume, simly becomes: V = A 23 3 (t 4 t ) (4) where A 23 is the area of the triangle with nodes,2 and 3. A 23 = 2 X X 2 X 3 Y Y 2 Y 3 In the generic case the Kinetic Energy Integral K n,n+ would take the form: K tet n,n+ = V 2 ρ (v n,n+ v n,n+ ) (42)

12 2 S. K. LAHIRI ET AL. But in the case of a tetrahedron with common nodes, the Kinetic Energy Integral K n,n+ would take the simle form (m 23 = ρ A 23 ): Kn,n+ tet = (t 4 t ) m (v n,n+ v n,n+ ) (43) Now, revisiting the sace-time-rism of the triangle (Fig. 2) it is observed, that it is subdivided into three tetrahedra, each one of them have a common node. Hence, using the above relations, a very simle form of the Kinetic Energy Integral is obtained: K rism n,n+ = m abc t 3 2 Where m abc is the mass of the triangle abc and: [(v a n+/2 va n+/2 ) + (vb n+/2 vb n+/2 ) + (vc n+/2 vc n+/2 ) ] t = t n+ t n v ai n+/2 = x ai n+ x n ai t a i = a, b, c Hence the kinetic energy integral for a generic rism for a corresonding triangular element e with mass m e can be written as: Kn,n+ e = me t ( ) vn+/2 a 3 2 va n+/2 (45) a=:3 Note, that the velocities used in each element is simly the nodal value. In case of a finite element mesh, the sace-time volume of the entire mesh can be subdivided into sace-timerisms corresonding to each triangular element: (44) K n,n+ (x n, x n+ ) = e K e n,n+ (46) The lumed mass of the node is obtained from the summation of all the elements linked at the node. For examle, the lumed mass at a node a in element e which has a global index, (reresented as: (e, a) ) M = (e,a) Hence, the net Kinetic Energy Integral obtained for the whole mesh would be: K n,n+ (x n, x n+ ) = Thus, the net Lagrangian of the entire mesh becomes: L n,n+ (x n, x n+ ) = m e 3 (47) t 2 M v n+/2 v n+/2 (48) t 2 M v n+/2 v n+/2 t Π(x n ) (49) This leads to the discrete Lagrangian Integral of the Central Difference method, as discussed in [59, 6]. Now the Kinetic Energy Integral is revisited to evaluate the directional derivatives: K n,n+ (x n, x n+ ) = t 2 M v n+/2 v n+/2 (5)

13 VARIATIONAL MESH ADAPTATION 3 where the velocity v n+/2 of node, can be written as: v n+/2 = t ( x n+ ) x n K n,n+ x n = M v n+/2 (5) Similarly : K n,n x n = M v n /2 (52) Using equation 7, we obtain the final discrete equation of motion as: K n,n+ x n + K n,n x n = M ( v n+/2 + v n /2 t Πint n x n ) t Πext n x n t (T n F n ) = (53) which can be simlified as: M ( v n+/2 v n /2 ) = t (F n T n ) (54) Thus, we obtain the time integration algorithm of Central Difference Scheme with lumed mass, using linear sace-time discretization. The urose of deriving the commonly known central difference method, was to demonstrate that it belongs to the class of variational integrators []. We have roved that the well known lumed mass aroximation is a consequence of linear sace-time discretization in case of linear triangular elements. In the rocess of deriving the method, we have also elucidated the use of sace-time discretization, which shall be used to develo time-integration udates for time-stes involving mesh changes in the next sections. 3. Mesh Adatation In this section, the reviously mentioned variational formulation, is extended to mesh adatation. Mesh adatations which involve local mesh changes for 2D triangular meshes, are considered. The following oerations are formulated searately:. Diagonal Swaing. 2. Node Movement. 3. Edge Slitting. 4. Edge Collasing. Each of these oerations is develoed with the assumtion that only one of these oerations takes lace between time level n and n + on a local atch.

14 4 S. K. LAHIRI ET AL. 3.. Diagonal Swaing A discussion of diagonal swaing is resented, by studying a local atch of two triangular elements abc and acd at time level t n, as shown in Fig. 3. The atch is time marched to time level t n+ where the common diagonal ac is swaed with the new diagonal bd, thus leading to two different element configurations, abd and bcd at time level t n+. The sacetime volume thus formed, can be subdivided into five tetrahedra: (a n b n c n b n+ ), (a n c n d n d n+ ), (a n+ b n+ d n+ a n ), (b n+ c n+ d n+ c n ) and (a n c n b n+ d n+ ) as shown in the figure 3. Note that, d t n+ a b c d c a t n b Figure 3. The sace-time volume for the diagonal swaing. the first four tetrahedra, have common nodes, hence the velocity interolation is simle. The velocity in the fifth(central) tetrahedra is comuted by the full exression, (as exlained in section 2). Hence, the net Kinetic Energy Integral within the sace-time volume can be written as: n,n+ = t m abc 2 3 vb n+/2 vb n+/2 + t m acd 2 3 vd n+/2 vd n+/2 + t 2 + t m bcd 2 3 vc n+/2 vc n+/2 + t m abcd vn+/2 abcd 2 3 vabcd ( macd xn+ b + m abc xn+ d m bcd xn a + m abd xn) c K abcd v abcd n+/2 = m abd 3 va n+/2 va n+/2 n+/2 (55) tm abcd (56)

15 VARIATIONAL MESH ADAPTATION 5 where m abcd = m abc + m acd. Using stationarity wrt. x n, the contribution to the inertial art of the equilibrium equations at t n arising from the rism abcd is: m abc D Kn,n+ abcd [δx n] = 3 vb n+/2 δx n b + m acd 3 vd n+/2 δx n d ( mabd + 3 va n+/2 + m ) bcd 3 vabcd n+/2 δxn a ( mbcd + 3 vc n+/2 + m ) abd 3 vabcd n+/2 δxn c (57) Adding this contribution to those arising from non-swaed elements in the mesh leads to an udate algorithm at ste t n which for nodes b and d is simly: ( ) Mn b vn+/2 b vb n /2 = t ( Fn b Tn b ) (58) ( ) Mn d vn+/2 d vd n /2 = t ( Fn d T n d ) (59) Note that as soon as the osition of nodes b and d have been udated, using equations 58 and 59, it is ossible to calculate v abcd n+/2 using 56 which in turns allows the udate of a and c to take lace as: M a n+v a n+/2 Ma nv a n /2 + m bcd 3 vabcd n+/2 = t (F a n T a n) (6) M c n+ vc n+/2 Mc n vc n /2 + m abd 3 vabcd n+/2 = t (F c n T c n ) (6) Similarly, using stationarity wrt. x n+, the contribution to the equilibrium relations at t n+ is obtained as: D 2 Kn,n+ abcd [δx n+] = m abd 3 va n+/2 δx n+ a + m bcd 3 vc n+/2 δx n+ c ( mabc + 3 vb n+/2 + m ) acd 3 vabcd n+/2 δxn+ b ( macd + 3 vd n+/2 + m ) abc 3 vabcd n+/2 δxn+ d (62) which lead to an udate algorithm at t n+ as: ( ) Mn+ a vn+3/2 a va n+/2 = t ( Fn+ a Tn+ a ) (63) ( ) Mn+ c vn+3/2 c vc n+/2 = t ( Fn+ c T n+ c ) (64) Mn+ b vb n+3/2 Mb n vb n+/2 m acd 3 vn+/2 abcd = t ( Fn+ b T n+ b ) Mn+ d vd n+3/2 Md n vd n+/2 m abc abcd = t ( Fn+ d T n+ d ) 3 vn+/2 The momentum within time ste t n and t n+ is (D 2 L n,n+ ): (65) (66) P n,n+ = P j n,n+ (67) j M j n+ vj n+/2 for j = a or c, P j n,n+ = Mn j vj n+/2 + ( m acd 3 )vn+/2 abcd for j = b, (68) Mnv j j n+/2 + ( m abc 3 )vn+/2 abcd for j = d. H n,n+ = j x j n+ Pj n,n+ (69)

16 6 S. K. LAHIRI ET AL Edge Slitting Now another atch of elements as shown in Fig. 4, is considered to develo the algorithm for edge-slitting. As shown in the figure, a atch of two triangles, abd and bcd, at time level t n, is time marched to time level t n+. The common edge bd is slit at midoint e to form d t n+ a e c d b c a b t n Figure 4. The sace-time volume for edge slitting. four child elements, abe, aed, bce and ecd, at time level t n+. The sace-time volume is now subdivided to five tetrahedra: (a n b n d n a n+ ), (b n c n d n c n+ ), (a n c n d n d n+ ), (a n c n b n b n+ ), and (b n d n a n+ c n+ ). Note that the first four tetrahedra have common nodes, hence the velocity interolation is simle. The fifth tetrahedra, is further subdivided into four tetrahedra (as shown in red dotted lines in Fig. 4), each having a common node as e. The oint e and the mid oint of b and d have the same reference coordinates, (X e = X b+x d 2 ). Thus the Kinetic

17 VARIATIONAL MESH ADAPTATION 7 Energy Integral can be written as: Kn,n+ abcd = t m abd 2 3 va n+/2 va n+/2 + t m bcd 2 + t m abe + m bce vn+/2 b 2 3 vb n+/2 + t 2 3 vc n+/2 vc n+/2 m dae + m dec vn+/2 d 3 vd n+/2 + t 2 m ev e n+/2 ve n+/2 (7) m e = (m abd + m bcd ) [ 3 ( ] x b n + xn) d v e n+/2 = t x e n+ Using stationarity wrt. x n one can obtain: D K abcd n,n+ [δx n] = 2 m abd 3 va n+/2 δx a n + m bcd 3 vc n+/2 δx c n + m abe + m bce vn+/2 b 3 δx n b + m dae + m dec vn+/2 d 3 δx n d + m e 2 ve n+/2 δx n b + m e 2 ve n+/2 δx n d (73) Thus, the udate algorithm at ste t n : ( ) Mn a vn+/2 a va n /2 = t (Fn a T n a ) (74) ( ) Mn c vn+/2 c vc n /2 = t (Fn c Tn) c (75) Mn+ b vb n+/2 Mb n vb n /2 m e 2 ve n+/2 = t (Fn a T n a ) (76) Mn+ d vd n+/2 Md n vd n /2 m e 2 ve n+/2 = t ( Fn d T n d ) (77) Due to the choice of the mid oint on the edge and linear elements, the osition of the new node at time level n can be assigned as xn e = x n b +x n d 2. Thus the velocity of the new node (also the velocity of the tetrahedron acbd) becomes : vn+/2 e = ( ) vn+/2 b 2 + vd n+/2 (78) Thus a 2 2 system of equation is obtained, to be solved, to obtain the other velocities. [ M b n+ + 4 m e 4 m e 4 m e M d n+ + 4 m e ] [ v b n+/2 v d n+/2 ] = [ M b n v b n+/2 M d n vd n+/2 ] + t [ F b n T b n F d n T d n The udate at equations at n+ are unchanged. The momentum within time ste t n and t n+ is (D 2 L n,n+ ): P n,n+ = j P j n,n+ = M j n+ vj n+/2 j ] (7) (72) (79) P j n,n+ (8) H n,n+ = j x j n+ Pj n,n+ (8)

18 8 S. K. LAHIRI ET AL Node Movement In order to derive the udate equations for node movement, the maing of the resent (satial) configuration to the reference (material) configuration, is revisited. An arbitrary intermediate configuration(ξ, η) is introduced, as shown in Fig. 5, as is tyically done in the case of Arbitrary Lagrangian and Eulerian formulation. The relations between the true and the observed velocity Y φ y X ψ η ϕ = φ o ψ x ϕ Figure 5. Understanding node movement with an intermediate maing. fields can then be written in the following manner: true velocity : v = x t = φ(x,t); (82) X=const. t observed velocity : ν = x t = ϕ(ξ, t); (83) ξ=const. t mesh velocity : V = X t = ψ(ξ, t); (84) ξ=const. t The true velocity v can be related to the observed velocity ν in terms of the mesh velocity V, using the deformation gradient: ξ dx = x dt t + x dx X dt ν = v + FV (85)

19 VARIATIONAL MESH ADAPTATION 9 The Kinetic Energy can then be written as: K n,n+ (x n,x n+ ) = t 2 M n+/2 v n,n+ v n,n+ (86) where: M n+/2 = ( M 2 n + Mn+) (87) v n,n+ = ν n+/2 F n V n+/2 (88) ν n+/2 = ( x n+ t x n ) (89) V n+/2 = ( X n+ t X n ) (9) ( ) Fn V = NnF n dv V NndV (9) The deformation gradient F n as used in equation 88 is evaluated at time level n in order to make the udate exlicit. The corresonding equilibrium equations are (for any generic node Figure 6. Sace-time volume for node movement., and its neighboring nodes s): M n+/2 v n,n+ M n /2 v n,n = tq n,n+ + t (F n T n) (92)

20 2 S. K. LAHIRI ET AL. where: v n,n+ = t Q n,n+ = a=,s [( x n+ x n ) ( F n X n+ X n )] m a n+/2 m a n (93) ( v a n,n+ Vn,n+ a ) ρ Nn a Nn dv (94) V In the case where neighboring nodes, s, remain fixed i.e. (V s n,n+ = ) and the total atch volume V remains fixed (imlying m n+/2 = ), the exression for Q m n n,n+ becomes: Q n,n+ = ( v n,n+ V ) n,n+ ρ Nn Nn dv = V Since the integral V ρ Nn Nn dv becomes zero for an internal node, for uniform density. Thus the udate ste for the node to be moved () becomes: M n+/2 v n,n+ M n /2 v n,n = t (F n T n) (95) And for the neighboring nodes (s), the udate ste becomes: Mn+/2 s vs n,n+ Mn /2 s vs n,n = tqn,n+ s + t (Fn s Tn) s (96) Qn,n+ s = ( v n,n+ V ) n,n+ ρ Nn Nn s dv (97) V The momentum within time ste t n and t n+ is (D 2 L n,n+ ): P n,n+ = j P j n,n+ (98) P j n,n+ = M n+/2 j v j n+/2 j H n,n+ = j x j n+ Pj n,n+ (99) 3.4. Edge Collasing Edge collasing oeration is aroached by visualizing a generic atch of elements, as shown in Fig. 7. In the triangular element arb, the edge ab is wished to be collased, leading to removal of the triangles arb and abs. The oints a and b, belonging to time level n, is substituted by the new oint c at time level n + as shown. The sace-time volume as shown in Fig. 8 is the volume over which the Lagrangian is to be comuted. To do so, the sace-time volume is sub-divided into tetrahedra. There are mainly three tyes of tetrahedra as shown in Fig. 9. The first tye (I) encloses the volume arbsc. Then based on the surrounding nodes there are two tyes of tetrahedra, as shown in Fig. 9. The tetrahedra having a or b as one of the vertices and the surrounding nodes from time level n and n +, are labeled tye (II). The tetrahedra having c as one of their vertices and the surrounding nodes from the time level n + and a or b as the fourth node, are labeled tye (III). The location of the new node c is chosen to be a linear interolation of the locations of nodes a,b and r. X c n+ = ξx a n + ηx b n + ( ξ η)x r n () x c n = ξx a n + ηx b n + ( ξ η)x r n ()

21 VARIATIONAL MESH ADAPTATION 2 4 s q 4 3 q 3 2 a c b q 2 r Figure 7. Collasing the edge ab to the oint c. q t n+ c s a b r t n Figure 8. The sace-time volume for Edge collasing oeration. The Kinetic Energy Integral and the velocity interolation within the tetrahedra of tye (I)

22 22 S. K. LAHIRI ET AL. Interior tetrahedra c s Tye I b a r Surrounding tetrahedra Tye III i+ c i+ c i a i a i+ i+ i i Tye II i+ a i Figure 9. The subdivision of the sace-time volume into different tyes of tetrahedra. can be written as follows: Kn,n+ I = t m n ab 2 3 vc n+/2 vc n+/2 (2) m n ab = m n arb + m n abs vn+/2 c = x n+ c (ξxn a + ηxn b + ( ξ η)xn) r (3) t Similarly the Kinetic Energy Integral and the velocity interolation within the tetrahedra of tye (II) can be written as: K II n,n+ = t 2 mn g i v gi n+/2 vgi n+/2 (4) v gi n+/2 = x gi n+ x n gi t (5)

23 VARIATIONAL MESH ADAPTATION 23 where the index g i is the overall index of all the neighboring nodes, ordered as (g i = r, q i, s, i ). The velocity in the tetrahedra of tye (III) is not straight forward, since there is no common node in each tetrahedron. Hence the full exression of the velocity (described reviously) is used. v gigi+ac n+/2 = A g i+acx gi n+ + A g iacx gi+ n+ + A g icg i+ xn+ c A g iag i+ xn a 6V gig i+ac (6) K gi(iii) n,n+ = m n g ig i+ac vgigi+ac n+/2 v gigi+ac n+/2 (7) This makes the algorithm very comlex. In order to simlify the algorithm, an aroximation is made. The Kinetic Energy Integral from each of the tetrahedra, of tye (III), are added together, and the sum is exressed by the following aroximation : g i K gi(iii) n,n+ = t 2 mn+ c vn+/2 v n+/2 (8) [ ] v n+/2 = tm n+ c m c = m n+ c mn ab 3 m c x c n+ + i m gi x gi n+ m a x a n m b x b n (9) () m gi = m n+ g i m n g i () m a = m n a mn ab 3 + M arcs (2) m b = m n b mn ab 3 + M bscr (3) Where M arcs and M bscr are the masses enclosed within arcs and bscr resectively. Note that m gi, m a and m b can be exressed as linear functions of ξ and η. The velocity v n+/2 is a weighted average of the velocities of all the tetrahedra of tye III as calculated in equation 6. In addition, since the neighboring nodes are not moved, nor are any neighboring edge allowed to be collased, the mass m n+ c is known ariori. Thus the net Kinetic Energy integral becomes: K n,n+ = Kn,n+ I + KII n,n+ + KIII n,n+ = t m n ab 2 3 vc n+/2 vc n+/2 + t 2 mn+ c vn+/2 v n+/2 + t 2 mn g i v gi n+/2 vgi n+/2 (4) i Using stationarity wrt. x n, the equilibrium equations obtained at time level n are as follows: ( g i r) m n g i v gi n+/2 mn g i v gi n /2 = t (F gi n T gi n ) (5) ( ξ η) mn ab 3 ξ mn ab 3 η mn ab 3 v c n+/2 + mn r vr n+/2 mn r vr n /2 = t (F r n T r n ) (6) v c n+/2 + m a v n+/2 mn a v a n /2 = t (F a n T a n) (7) v c n+/2 + m b v n+/2 mn b vb n /2 = t ( Fn b T n b ) (8)

24 24 S. K. LAHIRI ET AL. Here, a new variable Rn j is introduced, where ( j = g i, a, b) Rn j = m n j v j n /2 + t ( Fn j Tn j ) (9) Note that Rn j is known ariori. Hence the set of equations, can be rewritten as: ( ξ η) mn ab 3 ( g i r) m n g i v gi n+/2 = R gi n (2) ξ mn ab 3 η mn ab 3 v c n+/2 + mn r vr n+/2 = R r n (2) v c n+/2 + m a v n+/2 = R a n (22) v c n+/2 + m b v n+/2 = R b n (23) Note here that Eqn. 2 is fully exlicit, hence, x gi n+ for all g i excet r are known. Now revisiting Eqn. 9 one can rewrite the exression for vn+/2 using Eqn. 2 as: where, vn+/2 = S m (ξ, η)vn+/2 c + W (ξ, η) (24) [ ] W (ξ, η) = tm n+ m gi z gi n+ m axn a m bxn b + m cxn c + m r xn r c i = W + ξ W ξ + η W η (25) z gi n+ = x gi n+ ( g i r) S m (ξ, η) = = t Rr n m r n m n ab 3m c n+ mr n ( g i = r) [ 3m c m r ] n ( m r ( ξ η)) m n ab = S + Sξ ξ + S η η + S2 ξ ξ2 + Sη 2 η2 + Sξη 2 ξη (26) Note that the vector coefficients (W,W ξ,w η ) and the scalar coefficients (S, S ξ, S η, S 2 ξ, S2 η, S 2 ξη ) are known ariori. Using Eqn. 24 in Eqns. 22 & 23 the two equations are rewritten as: where, K a (ξ, η) v c n+/2 + m a (ξ, η) W (ξ, η) = Ra n (27) K b (ξ, η) v c n+/2 + m b (ξ, η) W (ξ, η) = Rb n (28) K a (ξ, η) = ξ mn ab 3 + m a(ξ, η) S m (ξ, η) (29) K b (ξ, η) = η mn ab 3 + m b (ξ, η) S m(ξ, η) (3) Now, eliminating vn+/2 c from both the above equations, the following equations are obtained: v c n+/2 = Ra n m a(ξ, η) W (ξ, η) K a (ξ, η) f (ξ, η) K b K a (R a n m a W ) + ( m b W R b n ) (3) = (32)

25 VARIATIONAL MESH ADAPTATION 25 Thus, a simle vector equation (32) is obtained, which is used to determine the scalars ξ and η by which, the osition of the new node c is determined. This is a couled quadratic equation which is solved by iteration. A simle Newton iteration leads to quadratic convergence. This leads to the osition of the new node (Xn+ c ) to be a solution of the local equilibrium. Edge ab is collased only if the node c lies within the area included by all the surrounding nodes g i. Once the osition of the node c is obtained, the velocity udates are obtained through simle exlicit equations mentioned above (3,24 and 2). The osition udates are obtained by the Eqns. 3 & 5. The momentum conserved in this time-ste is of the form: P n,n+ = j P j n,n+ (33) P j n,n+ = H n,n+ = j { m j n v j n+/2 + m jvn+/2, for j = g i 3 mn ab vj n+/2 + (mj n+ mn ab 3 )v n+/2, for j = c (34) x j n+ Pj n,n+ (35) Similar to the revious time-ste, using stationarity wrt. x n+, the equilibrium equations for the next time ste t n+ are obtained. The final udate equations are: m gi n+ vgi n+/2 mgi n vgi n+/2 m g i v n+/2 = t (F gi n+ T gi n+ ) (36) m c n+v c n+/2 3 mn abv c n+/2 (mc n+ mn ab 3 )v n+/2 = t (F c n+ T c n+) (37) 4. Error Estimate and Adatation Criteria Using the mesh adatation rocedures, exlained so far, an effective mesh-adative solver can be imlemented which is momentum conserving. In order to develo a mesh adative solver, a suitable mesh adatation criteria, based on error estimates was used. A gradient-tye error estimate described by Zienkiewicz and Zhu,[46, 47] (commonly known as Z 2 error-estimate) was used. The stresses in each element, and its neighboring elements, were used to obtain a recovered stress at the element. The difference of these two stresses leads to the error estimate at the element. The details of this error estimate can be found in [59]. Elements with high values of this error, were chosen for adatation. Mesh adatation was erformed as a sequence of all the mesh oerations (diagonal swaing, edge slitting, edge collasing or node-movement) in tandem. Each mesh oeration involved two timestes (t n t n+ ) and (t n+ t n+2 ). Only one tye of mesh oeration was attemted within each air of timestes, over the whole mesh. Mesh oerations were attemted after constant intervals (number of time-stes). At the beginning of each mesh oeration, a simle non-adative udate was conducted. Then based on the error-estimate, elements were chosen. Using the element connectivity information, a local atch, associated with the chosen element was identified. Any other element touching this atch (sharing a common node) was restricted from that mesh-oeration in that timeste. ( Within the element, edge-length-ratio of each edge (η i ) was obtained by the relation η i = P 2li, l i being the length of the i th edge. The edge-length-ratios varied from to. j lj )

26 26 S. K. LAHIRI ET AL. Values close to zero or one, indicated distorted elements. Edges, with such extreme values of edge-length-ratios, were collased, slit or swaed. Edges with edge-length-ratios close to, were collased. Edges with edge-length-ratios close to, were swaed or slit. The minimum edge-length-ratio for collasing or the maximum edge-length-ratio for swaing or slitting were threshold values chosen for each roblem. In case of node movement, a local atch of nodes were considered, and the average (centroid) location of the nodes and the deviation of the node from the average location was calculated. For higher deviation values, the node was moved towards the centroid. Figure. Mesh adatation oerations conducted in tandem, each oeration requiring a air of timestes. Diagonal swaing, node movement, edge-slitting and edge-collasing were attemted in this sequence at every subsequent (or alternate) timeste air, as shown in figure. Based on resolution requirements, the lowest (finest) hierarchical level of the grid was rescribed, in order to revent over-refinement. The zero th (coarsest) hierarchical level elements were not removed, in order to revent over-coarsening. Tyically the threshold values for edge-length ratios, maximum hierarchical levels and the frequency of adatation were secified as inut arameters for each oeration. The adatation criteria used for the resent mesh oerations were chosen for their simlicity of imlementation. Further develoment of the mesh adatation criteria, is required for generic cases. In the next section, erformance of the final mesh-adative solver is demonstrated using examles from raid dynamics of hyerelastic bodies. The urose behind these examles is to demonstrate that the udates develoed in the revious sections are momentum conserving. Such mesh adative time udates can be imlemented with different mesh-adatation criteria, with very less additional comutational exense. Time integration was erformed such that the entire mesh was ket at the same time level (synchronous integration). The critical characteristic mesh size, h, was chosen as the smallest radius of the inscribed circle in each element. The time ste was calculated based on the characteristic mesh size, maximum wave seed ( v max = v +a, with a being the fastest wave seed in the material and v is the maximum velocity in the mesh ) and the CFL (Courant-

27 VARIATIONAL MESH ADAPTATION 27 Friedrichs-Lewy) number, ν, as: t = ν h v max (38) CFL number s tyically ranged for. to.4 for the following examles. Lower CFL numbers are commonly encountered in more comlex raid dynamics roblems. 5.. Sinning Plate 5. Examles A unit thickness square late, sinning without any constraint, was considered as a test case to illustrate the conservation roerties of the roosed mesh adatation rocedures. The late was made out of nearly incomressible rubber material with material roerties, viz., Young s Modulus E =.7 7 Pa, Poisson s ratio ν =.45 and density ρ =. 3 kg/m 3. The late rotated at RPM. The late was meshed with 2 equal linear triangular e+4-2.e+5-3.4e+5-4.8e+5-6.2e+5-7.6e+5-9.e+5 -.4E+6 -.8E E E+6 -.6E E E+4 -.7E+5-2.8E+5-3.9E+5-5.E+5-6.E+5-7.2E+5-8.3E+5-9.4E+5 -.5E+6 -.6E E E+6 y.4 y x x Figure. Examle 5. : A Sinning late simulation with adatation. elements as shown in Figure (left) which also shows the ressure distribution at a given instant (right). The simulation was conducted using a mesh-adative solver, using all the above mentioned, adatation rocedures viz., diagonal swaing, node movement, edge-slitting and edge collasing as exlained in the revious section. Figure, demonstrates how the mesh was refined where the mesh skewness and stresses were relatively larger. The linear momentum P n,n+ and angular momentum H n,n+ are exected to remain constant during the time integration. The center of mass was initially at X =.5,Y =.5, and as a consequence of conservation of linear momentum P n,n+, remains at the same location. The momentum remains conserved exactly, throughout the simulation as shown in figure 2. A zoomed in lot deicts the variations of momentum at a smaller scale, which are of the order of the machine recision and remains bounded over the entire simulation. The momentum

28 28 S. K. LAHIRI ET AL. 25 X_Mom. Y_Mom. Ang. Mom. 2E- x momentum y momentum 2 E- 5 Momentum 5 Momentum -E- -2E Time time Figure 2. Examle 5.: Linear and angular momentum history. The figure at the right shows the same lot of linear momentum at a smaller scale. calculated in each ste was based on the P n,n+ and H n,n+ exressions described in each of the adatation rocedures reviously. The center of mass remains at (.5,.5) as shown in figure.2 x_cm y_cm.8 Location of center of mass Time Figure 3. Examle 5. : Location of center of mass. 3. Viscous stabilization as described in [59], was used to dissiate error accumulation over several time integration stes. Energy history, as shown in figure 4, shows very less energy dissiation in this case. Figure 4, also shows the adatation history, with the cumulative number of oerations conducted and the total number of nodes and elements at each instance of time during the course of adatation. Figure 5 shows the behavior of the error (as comared to the error in the finer mesh) with mesh size and the relative comutational time involved. The mesh size for the adated solution was taken as the mean of the coarsest and finest mesh

29 VARIATIONAL MESH ADAPTATION 29.2 Kinetic Energy Potential Energy Total Energy Swa Move Slit Collase Nodes Elements Energy Number of adatations Time time stes Figure 4. Examle 5. : Energy history (left) and Adatation History (right). Figure 5. Examle 5. : The figure in the left shows the error behavior with mesh size. The figure in right shows the comutational times for each of the comutations. size within the adated solution. It is shown through figure 5 (right), that error-reduction is accomlished using mesh-adatation with low additional comutational costs An oscillating ring A unit thickness circular ring, made u of nearly incomressible hyerelastic material (rubber) (E =.7 7 Pa, ν =.48 and ρ = 3 kg/m 3 ) is initially stretched to.5 its diameter and thereafter let to oscillate freely. This was chosen as another test case to study the momentum conservation roerty of mesh adatation. The initial configuration of the ring is shown in figure 6 (left). The ring was stretched as shown in figure 6 (right), at time t = and thereafter let to oscillate freely. The simulation was erformed u to time t =.2s, involving 244 time stes, with mesh adatations at every stes. Figure 7-8 show the satial (deformed) configurations at intermediate time stes. Linear and Angular momentum remains conserved throughout the simulation. The center of mass

30 3 S. K. LAHIRI ET AL. y.5 -.9E+7 -.9E E E E E E E E E E E+7-2.E+7 y E+7 -.9E E E E E E E E E E E+7-2.E x x Figure 6. Examle 5.2 : Ring at time t = s, with material(left) and satial(right) configurations. Figure 7. Examle 5.2 : Ring at time t =.5 s (left) and t =. s (right) Figure 8. Examle 5.2 : Ring at time t =.5 s (left) and t =.2 s (right)