A Total Lagrangian Based Method for Recovering the Undeformed Configuration in Finite Elasticity

Transcription

1 A Total Lagrangian Based Method or Recovering the Undeormed Coniguration in Finite Elasticity Grand Roman Joldes*, Adam Wittek and Karol Miller Intelligent Systems or Medicine Laboratory, School o Mechanical and Chemical Engineering, The University o Western Australia, 35 Stirling Highway, Crawley/Perth WA 69, AUSTRALIA ABSTRACT The problem o inding the un-deormed coniguration o an elastic body, when the deormed coniguration and the loads are known, occurs in many engineering applications. Standard solution methods or such problems include conservation laws based on Eshelby s energy-momentum tensor and re-parameterization o the standard equilibrium equations. In this paper we present a dierent method or solving such problems, based on a re-parameterization o the nodal orces using the Total Lagrangian ormulation. The obtained nonlinear system o equations describing equilibrium can be solved using either Newton-Raphson or an explicit dynamic relaxation algorithm. The solution method requires only minor modiications to similar algorithms designed or orward motion calculations. Several examples involving large deormations and dierent boundary conditions and loads are presented. KEYWORDS: un-deormed coniguration; Total Lagrangian ormulation; explicit time integration; dynamic relaxation; Newton-Raphson iterations * Corresponding author. Tel: , Fax: grand.joldes@uwa.edu.au 1

2 1. INTRODUCTION The problem o inding the un-deormed shape o an elastic body, when the inal deormed shape, material behavior, applied loads and boundary conditions are known, is encountered in many dierent engineering applications. A classic example is the design o rubber seals, which are pressed into a channel and need to exert a desired pressure onto the channel. Another application is the design o rubber orms to be used in pressing thin sheets o metal in stamping procedures [1]. This problem stated above has been called by some authors an inverse elastostatics problem [2-4]. As many elasticity problems related to model parameter identiication are also called inverse problems, in order to avoid conusion we will use the term direct deormation when reerring to the computation o deormation in an elastic body under known loads starting rom the un-deormed coniguration and the term inverse deormation when reerring to the computation o deormations in an elastic body under known loads starting rom the deormed coniguration. More recently the need or solutions to inverse deormation problems was also identiied in various areas o bio-engineering. In [5] an application to breast biomechanics is presented. The imaging o the breast is perormed with the patient lying in a prone position (on the stomach). Surgical procedures such as breast biopsy are usually perormed with the patients lying on their back (supine position). The reerence un-deormed state o the breast is needed in order to predict the breast shape and perorm image registration to the new patient position. In [2, 3, 6, 7] the solution o an inverse deormation problem is used in order to assess stresses in the walls o aneurysms. Sometimes the solution to an inverse deormation problem is avoided by modelling the structure in such a way that the stresses can be computed directly based on the geometry and applied loads. For example, aneurysms are modelled as membranes in [8]. A solution method or inverse deormation problems in inite elasticity was proposed in [9], by ormulating the problem as a set o balance equations written in terms o inverse deormation and standard boundary conditions. This ormulation exploits a set o duality relations that allow the ormulation o an inverse deormation problem in a 2

3 orm that appears similar to a standard elastostatic problem. Later, Chadwick [1] reormulated the equilibrium equations in terms o Eshelby s energy-momentum tensor [11]. In [4] Govindjee and Mihalic show that the energy-momentum ormulation has several deiciencies: it places strong continuity requirements on the motion, Eshelby s tensor lacks direct physical connection to the stated problem creating diiculties with the boundary conditions and it cannot handle body orces. They propose a new method based on the re-parameterization o the equilibrium equations, which eliminates these diiculties. The method is later extended to incompressible materials in [1], and was shown to be consistent with the approach presented in [12]. The methods presented so ar require the writing o the stress equilibrium equations in terms o the deormed coniguration (Eulerian description). Conventional orward inite elasticity analyses are typically ormulated with respect to a Lagrangian rame o reerence (based on the un-deormed coniguration) [13]. Rajagopal et. al present a method o solving inverse deormation problems using a Lagrangian rame o reerence in [5]. Their method is based on a inite dierence approximation o the Jacobian or the system o equations describing the equilibrium, considering the parameters o the deormed state as an initial estimate or the parameters o the reerence state. In [7] Riveros et. al propose a method which uses the displacements obtained by solving the direct problem using the current coniguration to iteratively update the geometry, which should converge towards the un-deormed geometry. In this paper we present a new method o solving inverse deormation problems. The starting point or our method consists o the standard equilibrium equations discretised using a Total Lagrangian (TL) ramework. The direct elastostatic problem can be stated, ater discretisation, as an equation deining the equilibrium o orces with the displacements as unknowns. The TL ramework allows a direct relationship between the un-deormed and the deormed conigurations to be deined. We exploit this relationship to rewrite the orce equilibrium equation based on the known deormed coniguration. The obtained non-linear system o equations can be solved using Newton-Raphson techniques, and we present the relations needed or the construction o the exact stiness 3

4 matrix required. We also present a dierent solution method based on explicit time integration [14] and dynamic relaxation [15]. The paper is organized as ollows: the derivation o the proposed method is presented in Section 2, several examples involving dierent boundary conditions and loads are presented in Section 3, ollowed by discussion and conclusions in Section PROBLEM FORMULATION AND SOLUTION METHOD 2.1. Derivation o equilibrium equation We consider hyperelastic materials or which the internal stresses and strains depend only on the deormation ield within the material and are independent o the way the deormation was applied (path-independent). For such materials the constitutive behavior is usually deined using a strain energy potential unction: ( I, I, J,...) (1) t W 1 2 where I 1 and I 2 are the irst and second strain invariants, computed based on the stretch matrix (Cauchy-Green strain tensor) and J is the total volume change, computed as the determinant o the deormation gradient [13]. The let subscript identiies the starting coniguration (in this case the un-deormed state, ) and the let superscript identiies the current coniguration. For anisotropic materials there may be other parameters describing the anisotropic behavior (such as the direction o ibers). Our derivation is based on the isotropic case, but can be easily extended to handle anisotropy. computed as: Based on the strain energy potential the second Piola-Kirchho stress, S, can be t S( t t W F) = t (2) E where E is the Green-Lagrange strain. Because both the strain energy potential and the Green-Lagrange strain are deined based on the deormation gradient F, the second Piola- Kirchho stress can be considered as a unction o the deormation gradient. The 4

5 deormed coniguration o an object can be characterized using the equilibrium between the internal orces in the inal deormed state and the externally applied orces: R int = R ext Ater the weak orm o the equilibrium equations is discretised (or example using a inite element discretisation), the deormation ield can be described using the nodes o the discretisation and their associated shape unctions. The internal orces can be expressed, ater discretisation using the Total Lagrangian ormulation, as [16, 17]: R int F T = D S T dv (4) V where F is the deormation gradient, V is the domain in the un-deormed coniguration and D is the matrix o shape unction derivatives in the un-deormed coniguration: H D x I only the inal coniguration and the corresponding external loads are known, the un-deormed coniguration (and the displacements) must be computed by solving the inverse deormation problem. We consider a discretization o the object in the inal coniguration using a set o n points x k. Using these points we can deine an interpolation unction to represent the displacements at any point in the object (using a inite element interpolant): n k T ( ) k ( ) k 1 u x H x u U H (6) I we consider the transormation that deines the transition rom the original to the deormed coniguration, φ:v V, the position o the discretizing nodes in the undeormed coniguration is given by: k 1 k ( ) (3) (5) x x (7) Based on these points we deine the shape unctions in the un-deormed coniguration as: H( x) H( x ) (8) The matrix o shape unction derivatives rom eq. 5 can thereore be expressed as: 5

6 D = H x = H x x x = H x F = D F (9) The positions o the points in the un-deormed coniguration o the object can be obtained by subtracting the deormations rom the inal positions, as: and, thereore, F = x x = ( x u) x x u (1) = I x = I U T H u = I x = I U T x D ( UT H) = x where I is the identity matrix. Based on the properties o the deormation gradient [13]: expressed as: F = (11) ( F) 1 = (I U T D) 1 (12) Finally, by using the change o variable (1) in eq. (4), the internal orces can be R int F T ) dv = x = D S T V = D S T V det ( x F T J 1 dv (13) Considering equations (2) and (9), equation (13) can be re-written as: R int = D F S T ( F) J 1 dv (14) V 6 F T with the deormation gradient computed based on the inal coniguration using eq. 12. In eq. 14 we re-wrote the internal orces based on the inal coniguration, which is known. We can now treat eq. 3, with the let hand side given by eq. 14, the same way as in the case o a direct problem, and ind the unknown displacements. I some o the loading (included in the right hand side o the equation) is dependent on the un-deormed coniguration, it must be re-written based on the deormed coniguration in a similar manner. For example, body orces (due to applied accelerations or gravity) or each element can be written as:

7 R b = H b T dv = H b T x det ( ) dv = H b T J 1 dv (15) V V x V Note 1: I the shape unctions H are linear over the elements o the discretization (and thereore the transormation φ, constructed using these shape unctions, is linear over the elements), considering the relation between the discretizing nodes in the deormed and un-deormed conigurations (eq. 7), the shape unctions H are also linear. Note 2: The exact numerical evaluation o the integrals that appear in equations 14 and 15 can be perormed only when constant strain elements with linear shape unctions are used. Using higher order shape unctions is also possible, but in such cases the unctions to be integrated are not polynomials and thereore exact integration is not possible by using simple quadrature rules. Given the above observation we will use constant strain elements (triangles/tetrahedrons with linear shape unctions) or solving inverse deormation problems. When almost incompressible materials need to be modeled, the non-locking tetrahedral element presented in [18] can be used Implicit solution methods The obtained equilibrium equation, which is based only on the known deormed coniguration, can be solved using standard solution algorithms, such as Newton- Raphson iterations. Existing solution methods can be easily modiied by replacing the subroutines or computing the stiness matrix based on the internal orces deined by eq. 4 with new subroutines based on the orces deined by eq. 14. The exact tangent stiness matrix based on the deinition o orces in eq. 14 should be used or optimum convergence o the Newton-Raphson iterations. We will present the equations needed or the derivation o the exact tangent stiness matrix in the ollowing paragraphs. as in [13]: The elements o the exact tangent stiness matrix or each element can be deined Rint mn K mnij = (16) U ij 7

8 where u ij is the displacement o node i o the element in the direction o coordinate j. In the ollowing equations we will omit the subscripts wherever possible. Thereore, considering eq. 14, we can re-write eq. 16 as: K = R int U = ( D F U ST F T J 1 ) dv (17) V The derivative under the integral in eq. 17 can be expanded as: U ( D F U ( F ST + F ST F T F T J 1 ) = S FT T U J 1 F U ) = D ( F U ST J 1 + F F T J 1 + ST F T ST U S F T J 1 U F T J 1 ) (18) F T J 1 + (19) We will now expand each o the derivatives that appear in eq. 19 (which deines, given the relationship between the Cauchy stress and the second Piola-Kirchho stress, the derivative o the Cauchy stress with respect to displacements). Based on the deinition o the deormation gradient, given by eq. 12, we can write: with F U F 1 = U 1 = F ( UT D U ) = F F 1 U F 1 = F 1 = F ( UT D U ) F (2) ( U T D) = U D i: e j (21) ij where D i: represents line i o matrix D, represents the outer product and e j is the versor corresponding to the degree o reedom j. The derivative o the second Piola-Kirchho stress matrix can be computed based on the ollowing equations: S U S E = E U where S represents the vector orm o the symmetric matrix S, and the Green-Lagrange strain tensor E is deined as (22) 8

9 The matrix E = 1 2 ( FT C F I) (23) S = E (24) describes the material behavior, and is the same as in the case o direct problems. The derivative o the Green-Lagrange strain tensor is E U = 1 2 FT ( U F + F T At last, the derivative o J -1 is calculated as: J 1 U = det ( F 1 ) = U det ( F) U = ( ( UT D U ) F = ( F U ) (25) U : F T : F T ) J 1 where : represents the inner product between matrices. ) det ( F) = (26) We implemented the above solution method in an existing 3D implicit solver or solving direct elasticity problems. The main change required is the modiication in the routine that computes the stiness matrix at element level the stiness matrix is computed based on the deinition o orces in eq. 14 (and the above presented eq ) instead o eq. 4. Another area that required changes is the storage o the stiness matrix, as or the inverse procedure the stiness matrix is no longer symmetric. This may also inluence the choice o linear solver (a solver specialized or symmetric matrices can no longer be used) Explicit solution methods Alternative algorithms or obtaining the steady state solution can also be used. By adding the acceleration term and a mass proportional damping term to the let hand side o equation 3, a standard dynamics equation o motion is obtained, which can be solved using explicit time integration combined with Dynamic Relaxation: M t x + cm t t t x + R int = R ext (27) 9

10 We modiied such an algorithm, presented previously by us in [15], developed or solving direct elasticity problems. The only modiication required was in the way the nodal orces or each element were computed. In the original code the nodal orces were computed using eq. 3. In the modiied code the computation was perormed by combining eq. 12 and 14. The resulting solution algorithm was used or solving many o the inverse deormation problems presented in the next section. The explicit integration algorithm used is only conditionally stable, with the stable integration step inluenced by the maximum eigenvalue o the stiness matrix. Because the modiied explicit algorithm uses a dierent expression or orce computation, the stiness matrix will also be dierent. Nevertheless, we were able to use the same element based critical time step estimation as or solving direct problems in most cases that did not include excessive element distortions. Because the purpose o a dynamic relaxation algorithm is to ind the steady state solution, the mass matrix is chosen only based on considerations o stability and convergence speed, as it does not inluence the steady state o the system [15]. Thereore, even i we start rom a deormed coniguration, there is no need to adapt the mass matrix to the changes in coniguration, and the same treatment as in the case o solving direct problems is used. For details on the derivation o the solution method, including adaptive selection o the parameters, we direct the reader to reerence [15]. The advantage o explicit methods over implicit ones depends on the size and type o the problem. A comparison between implicit and explicit methods, presented in [19], shows that the implicit method requires 4 times more computations per time steps when compared to the explicit method or 3D problems. Due to their low memory requirements, explicit methods become even more attractive or problems with very large number o degrees o reedom, where the stiness matrix cannot be stored in the ast core memory (RAM). The presented explicit solution algorithm is very well suited or parallel implementation on graphics processing units (GPU). We implemented the explicit inverse deormation solution method on GPU ollowing the same approach as or the direct solution method, due to the minimal dierences between methods (computation o nodal orces). The details o the implementation are presented in [2]. 1

11 3. EXAMPLES In the ollowing examples we used plane strain triangular elements or 2D simulations and constant strain tetrahedral elements or 3D simulations. We considered a Neo-Hookean material with the strain energy unction: K 2 t 2 W ( I1 3) ( J 1) (28) where I 1 is the irst invariant o the deviatoric Cauchy-Green strain tensor, μ is the initial shear modulus and K is the initial bulk modulus. For most simulations, unless otherwise stated, we used a sot material with an initial Young s modulus o 3 Pa and a Poisson s ratio o.49. The irst 4 examples were solved using the explicit inverse deormation procedure, while the last one was solved using the implicit inverse deormation procedure. The general setup o the numerical experiments is as ollows: we start with an initial known geometry, G, and loading. We then apply a direct procedure ollowed by the proposed inverse deormation procedure, or the inverse deormation procedure ollowed by a direct deormation procedure, and obtain a inal geometry, F. We assessed the accuracy o the results by comparing the positions o the nodes o the initial geometry, G, with the positions corresponding to the inal geometry, F. The dierence in nodal positions compounds the errors rom both the direct and the inverse deormation solutions. For perect solutions, there should be no dierence in nodal positions. Numerical methods cannot provide a perect solution, as there will always be errors generated by truncation, rounding, and convergence criteria used. Thereore, the inverse deormation solutions are proven to be correct as long as the obtained dierences in nodal positions are deemed small. This is shown to be the case in all the ollowing examples. 11

12 For the problems solved using the explicit method we used a large number o time steps in order to reduce the inluence o the convergence criteria on the accuracy o the numerical solution. 3.1 Extension and compression o a rectangle In this example we perormed a constrained extension and compression o a rectangular domain by applying essential boundary conditions (displacing one o the aces while constraining the opposite ace). We recovered the original shape o the rectangle using the explicit inverse deormation solution method, with the same essential boundary conditions. We used 2 time steps or load application and 3 time steps or inding the equilibrium position using Dynamic Relaxation, or both the direct and the inverse deormation solution algorithms. This example demonstrates the handling o very large deormations, with the rectangle being compressed by 35% o its length and extended by 5% o its length. The deormed and un-deormed conigurations are presented in Figure 1. The maximum obtained dierences in nodal positions between the original and inal geometries are presented in Table 1. Possible location o Figure 1. The un-deormed/recovered (a), compressed (b), and extended geometry (c). Table 1. Results rom the rectangle extension and compression experiments. Experiment Rectangle size [mm 2 ] Maximum applied deormation [mm] Dierence in nodal positions [mm] Compression 4x E-4 Extension 4x E-5 12

13 3.2 Gasket shape In this example we consider the desired shape o a gasket while compressed and the distribution o pressure required at the interace (Figure 2.a). We then calculate the shape that should be manuactured (uncompressed) or two cases. In the irst case there is high riction between the compressing surace and the gasket, simulated here, same as in [4], as a restriction on the lateral motion o the nodes ound in contact with the surace (Figure 2.b). In the second case there is no riction between the gasket and the compressing surace (Figure 2.c). Because o symmetry, only 1/4 o the gasket is actually modeled, and the symmetrical behavior is imposed by using suitable essential boundary conditions. This example demonstrates the correct handling o surace pressures and symmetries. We used 1 time steps or load application and 2 time steps or inding the equilibrium. The required pressure is set to 4 kpa at the middle o the gasket, increasing linearly to 12 kpa on the lateral quarter o the gasket. The resulting variation o the nodal orces at the interace is shown in Figure 2.a. The accuracy o the solution was checked by solving the direct problems and comparing the desired shape o the gasket with the computed one. For the direct problem solution the compression o the gasket was perormed by moving the compressing surace and using a simple kinematic contact algorithm that prevented the interacing nodes rom penetrating the surace. This way we can also compare the resulting nodal orces with the required ones. The results o the analysis are presented in Table 2. 13

14 Possible location o Figure 2. a) The deormed shape o the gasket (only 1/4 o the geometry shown, nodes that have essential boundary conditions applied due to symmetry are marked). The variation o the desired nodal orces on the compressed surace is also shown. b) The undeormed geometry with riction. c) The un-deormed geometry with no riction. Table 2. Results rom the manuacturing gasket shape experiments. Experiment Gasket size [mm 2 ] Dierence in nodal orces [N] Dierence in nodal positions [mm] Friction 3x E-6 No riction 3x E Beam under gravity loading The purpose o this example is to demonstrate the capacity o the method to handle body orces (see eq. 15). A rectangular beam is ixed at one end and subjected to gravity loading. The un-deormed shape is then recovered using the proposed algorithm. We used 4 time steps or load application and 6 time steps or inding the equilibrium. The maximum dierence in nodal positions between the un-deormed and the recovered geometries was.42mm or a beam o 4x1 mm 2. 14

15 Possible location o Figure 3. a) The deormed shape o the beam under gravity load; b) The undeormed/recovered geometry D object under gravity loading This example demonstrates that our inverse deormation procedure implementation, including handling o gravity orces (whose value varies with the volume o the object) accurately computes the un-deormed coniguration or 3D objects. It also demonstrates the eiciency o a parallel implementation o the explicit inverse deormation procedure on Graphics Processing Units (GPU). To assess the accuracy o the method we perormed the ollowing experiment: or a chosen 3D geometry, considered as the deormed coniguration under the eect o gravity, we applied our explicit inverse deormation procedure to compute the undeormed coniguration. The resulting un-deormed geometry was transerred into the commercial inite element sotware Abaqus [21], and loaded by gravity to compute the inal geometry. The direct implicit solver with the deault settings was used in Abaqus. We then compared the inal geometry obtained rom Abaqus with the starting deormed geometry or our inverse deormation procedure, deining the error as the dierence at each nodal position (Figure 4.b). For solving the direct problem we also used our explicit dynamic relaxation procedure. 15

16 Possible location o Figure 4. a) The un-deormed geometry o a 3D object. b) The deormed geometry under gravity loading. Colors mark the dierence in nodal positions between the geometry selected as the deormed geometry and the geometry recovered ater applying our inverse deormation procedure, ollowed by a direct procedure. The direct procedure used was our explicit dynamic relaxation procedure in a) and the Abaqus implicit procedure in b). 16

17 In order to increase the eect o gravity we considered a very sot and very heavy material, with a Poisson s ratio o.4. The resulting maximum displacement under gravity loading was 11% o the object s height. The maximum error relative to maximum displacements was.7% when the Abaqus implicit solver was used and.8% when our explicit dynamic relaxation solver was used. In terms o eiciency o the parallel implementation on GPU, we compared the speed o the direct explicit solution method to the speed o the inverse deormation explicit solution method. For this problem, the mesh consisted o approximately 575k elements and 1k nodes (3k degrees o reedom). The time required or the execution o 1 explicit steps on GPU was 12.2s or the inverse deormation method and 7.62s or the direct method. Both methods executed more than 15 times aster on GPU than on the CPU and required approximately 55 explicit steps to converge. We were unable to solve the problem using the implicit solution method due to insuicient RAM or storing the stiness matrix, as our implementation cannot handle out o RAM matrix storage. The direct solution computation using the implicit Newton- Raphson method took 4 iterations and 2239s in the inite element solver Abaqus, as compared to 63s required by the explicit solution on the CPU and less than 4s on the GPU. It is expected that an inverse deormation implicit solution method will take even longer to compute, due to the lack o symmetry o the stiness matrix. Computations were done on a PC with an Intel Core 2 Quad processor, 3 GB o RAM and Windows XP operating system. The GPU used was an NVIDIA Tesla C16 having 4 GB o RAM and 24 cores. The newer generation GPUs, such as Tesla K4, can have up to 288 cores, thereore promising even aster execution times. 3.5 Very large deormations in 3D This example presents the deormation o a semi-circular rubber band with a square section resulting rom the application o a constant orce to its middle section (Figure 5). This example showcases the use o implicit Newton-Raphson procedures or solving 3D problems involving very large deormations. 17

18 The rubber band was discretized using approximately 25k tetrahedral elements and 6k nodes. Due to the very large resulting deormation, we applied the load in 2 steps and used a line search algorithm to maintain the convergence o the Newton-Raphson procedure. The linear solver used was MUMPS [22], which can handle both symmetric positive deinite matrices (or direct problems) and general un-symmetric matrices (or inverse deormation problems). Possible location o Figure 5. Deormation o a rubber band under a orce applied to its middle section. The un-deormed shape is shown as a mesh. The colors o the deormed shape mark the size o the resulting nodal displacements. Possible location o Figure 6. Convergence in displacements or the direct (a) and inverse deormation (b) Newton-Raphson iterations. 18

19 We started with the un-deormed shape and applied the orce to ind the deormed shape. We then used the inverse deormation implicit procedure to recover the undeormed shape. The maximum dierence in nodal positions between the original and recovered un-deormed geometries was 1.1e-1. In terms o displacement convergence o Newton-Raphson iterations, or this particular problem, the inverse deormation procedure shows much aster convergence as compared to the direct procedure. For an error limit o 1e-3, the direct procedure converges in 46 iterations while the inverse deormation procedure converges in 9 iterations (Figure 6). The error or each iteration step is deined as the dierence between the displacements calculated ater that iteration and the displacements calculated ater a large number o iterations (a maximum o 3 iterations per load step was conigured). The use o the exact tangent stiness matrix ensures good convergence properties or implicit inverse deormation solution methods. 4. DISCUSSION AND CONCLUSSIONS In this paper we present a method or solving inverse deormation problems. The commonly used method or solving such problems involves the re-parameterization o the equilibrium equations in terms o the deormed coniguration (Eulerian description). Our method exploits the relationship between the un-deormed and the deormed conigurations provided by the TL ramework to rewrite the orce equilibrium equation based on the known deormed coniguration. The proposed method has several advantages over the commonly used method presented by Govindjee and Mihalic in [4], especially when implemented in existing inite element sotware: It does not require the computation o the tangent operator describing the derivatives o the Cauchy stress to the inverse o the Cauchy-Green strain tensor (D = σ/c); existing material law implementations describing the relation between strain and stress can be used. 19

20 The main change required or existing TL inite element sotware or solving direct problems is in the computation o stiness matrix (implicit solution) or nodal orces (explicit solution) at element level. The explicit algorithm is very well suited or parallel implementation on Graphics Computing Units (GPU) [2, 23], which can lead to signiicant speed-ups or highly non-linear problems with a large number o degrees o reedom. We presented several examples o problems solved using the proposed method implemented in a TL explicit dynamics sotware which uses dynamic relaxation to ind the static equilibrium solution. The examples demonstrate that the presented method can solve the inverse deormation problems under diverse loads (displacements, surace tractions and body orces). Although we were able to obtain results using the standard procedures or estimating the critical time step, we must mention that these procedures may not always work. The Jacobian o the system o equations describing the equilibrium is dierent rom the one used in direct problems and it lacks symmetry (this is also the case or the method proposed in [4]). Computation o the critical time step or the inverse deormation explicit solution algorithm will require urther investigation. The presented explicit solution algorithm is very well suited or parallel implementation on graphics processing units (GPU). The presented example solved using our GPU implementation demonstrates the potential o greatly reducing the computation time or problems with a large number o degrees o reedom. We developed the relations or the derivation o the stiness matrix to be used with Newton-Raphson inverse deormation solution methods. We showed that such methods can be used to solve inverse deormation problems involving very large deormations at machine precision, thereore demonstrating the correctness o the presented approach. Acknowledgements: The inancial support o the Australian Research Council (Grant No. DP and DP12142) is grateully acknowledged. 2

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