Discontinuous Galerkin methods for nonlinear elasticity

Transcription

1 Discontinuous Galerkin methods for nonlinear elasticity Preprint submitted to lsevier Science 8 January 2008

2 The goal of this paper is to introduce Discontinuous Galerkin (DG) methods for nonlinear elasticity and demonstrate some of their advantages over more traditional methods. Although the use of DG for elliptic problems may not readily appear advantageous since we typically expect solutions to be smooth, we shall see that when the constraint of inter-element continuity is relaxed not only are some problems easier to solve, but, in some cases, we may be able to solve them more accurately. For instance, consider a nearly incompressible neo-hookean material. It is well known that low order conforming elements will lock and one must use either a mixed method or high order elements. As we will see, when continuity is relaxed, a DG displacement-only formulation with low order elements does not lock. To see an example more clearly, consider a very poorly chosen mesh such as in Fig. 1. Due to the structure, a conforming method will produce asymmetric deformations even when symmetric loads are applied. However, when we relax the continuity constraint, we see that DG does not suffer this pathology. In addition to reporting some of the advantages, we also hope to refute some of the common thought disadvantages of DG for elliptic problems. There is often a predominant concern over the excessive costs stemming from extra degrees of freedom (i.e. the discontinuities). We will see that in some cases, the DG approximation to the derivative is slightly more accurate than a highly-refined conforming approximation with the same number of degrees of freedom. This means that a given level of strain accuracy may be just as expensive for a conforming method as it is for DG. Another concern involves the loops over interior element faces which adds a significant cost in three-dimensions. As we will see, DG can be formulated such that no loops over interior faces are needed to assemble the linearized elasticity problem. In this case, the assembly routine is virtually the same as that of a conforming method with a slight modification to the element matrices. Another common concern is the resulting illconditioned linearized problem. This could make it impossible to solve large three-dimensional problems in parallel. We found that the traditional Additive Schwarz preconditioner seemed (a) DG approximation (b) Conforming approximation Fig. 1. The reference configuration is a compressible neo-hookean square with unit length. The upper surface is translated down via Dirichlet boundary conditions with 100 increments. The left figure shows an intermediate configuration, the right figure shows the final configuration. For the DG method we set β = 0.01k where k is the increment number. to work reasonably well for a worst case scenario in three-dimensions where DG is used to model artery micromechanics. The linear system of equations has 4.4(10 6 ) unknowns and is extremely ill-conditioned since the material coefficients rapidly oscillate by a factor of Despite this rather arduous challenge, we are able to solve the linearized system in parallel using 343 processors with a total solve time of 30 minutes. For many supercomputing clusters, this is not an unreasonable request. Another concern is that most DG methods are sensitive to the method of stabilization which almost always is a penalty on discontinuities. Penalizing discontinuities can cause problems, since these are the very degrees of freedom in which many of DG s benefits arise. Several authors have shown that it can be easy to over penalize and substantially degrade the performance of the DG method. We will see that DG can be stabilized in such a way that the performance is virtually unaffected by the user s choice of penalty. Based on our research thus far, we see no reason to discount DG as a viable tool to solve real world nonlinear elasticity problems. 2

3 Let us begin by defining the nonlinear elasticity problem for a body that occupies a connected open domain B 0 R d, with d = 2, 3. The body deforms due to the presence of applied forces or displacements on the surface, B 0. The deformation of the reference configuration is described by the deformation mapping, ϕ : B 0 R d. The mechanical behavior of the material is assumed to be hyperelastic and defined by a strain energy density W : R d d R, W (F), which depends on the deformation gradient, F(X) = ϕ(x). Mechanical loads in the form of surface tractions T are applied on τ B 0 B 0. The potential energy of the body when deformed with a deformation mapping ϕ is computed as I[ϕ] = W (F) dv T ϕ ds. (1) B 0 τb 0 The nonlinear elasticity problem consists in finding the deformations in which the body is in mechanical equilibrium under the imposed boundary conditions. We seek these configurations within a space of admissible deformations V. The set of all admissible variations of this space consists of all functions of the form δϕ = ϕ ɛ / ɛ ɛ=0, where ϕ ɛ is a smooth one-parameter family of functions in V. An equilibrium configuration of the body is a deformation mapping ϕ V that renders the potential energy stationary with respect to all admissible variations. For the discrete problem, we begin by defining Vh, a discrete vector space of smooth scalar functions on an element. The DG finite element space is then V h = Π Vh, and hence the elements of V h are allowed to have discontinuities across element boundaries. For the discrete nonlinear elasticity problem we shall use two types of such spaces, V h and W h. The DG space Vh d will be used to approximate the deformation mapping ϕ, and the space Wh d d will be used to approximate its gradient (rather than Vh d ). In order to define the discrete approximation to the gradient we use the lifting operator R : L 2 (Γ) d Wh d d which requires R(ϕ) z dv = B 0 Γ ϕ N {z } ds, (2) Traction (Pa) e-04 1e-05 1e-06 xact-ν=0.5 Conforming-ν= DG-ν= Conforming-ν=0.499 DG-ν= r 0 /R 0 Fig. 2. Tractions measured on the inner wall of an expanding cylinder as a function of the ratio of the deformed to reference radius using 10,000 increments (1.1 represents a 10% increase in radius). Stabilization parameter β = 10 4 k where k is the increment number. As measured, the traction represents the internal pressure needed to expand the cylinder to a given radius. Notice that for the conforming elements the tractions are several orders of magnitude larger than necessary. However, for DG we see a good approximation to the exact solution. for all z Wh d d, where Γ is the set of all element boundaries and {ϕ } = 1 2 (ϕ+ + ϕ ). The calculation of the lifting operator is the only operation that involves a loop over element faces. However, this is a preprocessing step that must be done once for a given mesh. Since its calculation involves a local element mass matrix inversion (and not a global one) the cost is trivial compared to solving the linearized elasticity problem. Once the lifting operator is computed the DG derivative D DG : Vh d W d d, is simply h D DG ϕ = ϕ + R([ϕ]) (3) where we have used the jump operator, [ϕ] = ϕ ϕ +. For this particular DG method (often called the Bassi-Rebay method) we clearly see a distinction, in the discrete derivative, between DG and conforming. The DG derivative uses the additional degrees of freedom (i.e. the discontinuities) via the lifting operator which takes functions in Γ and returns an enhancement in Wh d d. This DG method yields a particularly simple formulation for the discrete linear elasticity problem. In 3

4 the discrete problem we again look for stationary points in the total potential energy, however now in the discontinuous space, Vh d. Here, we use the same total potential energy as shown in q. 1 now approximating ϕ with ϕ h, and F with D DG ϕ h. The resulting problem is nonlinear, with the nonlinearity embedded in the constitutive relation, W (D DG ϕ h ). It can be solved by taking a second variation, δ 2, and using Newton type iteration. In the absence of external tractions, the linearized problem becomes W F (D DG ϕ h) : δd DG ϕ h dv = δd DG ϕ h : 2 W F 2 (D DG ϕ h): δ 2 D DG ϕ h dv. (4) Most DG methods will display numerical instabilities without proper stabilization. In nonlinear elasticity, numerical instabilities can be an especially challenging task due to the loss of positive definiteness of the elasticity tensor, A = 2 W/ F 2, in selected regions of B 0. Typically, a stable method is achieved through a penalty on the discontinuities. Using the traditional approach one would add to the total potential energy a term such as β [ϕ h h ] 2 ds, (5) Γ where β > 0 and h is a measure of the mesh size. In Fig. 2 we see an example where traditional stabilization is used to approximate the deformations of a nearly incompressible neo-hookean material. Despite having to satisfy the nonlinear constraint, det(f) = 1, we see that DG does not lock even though a purely displacement formulation is used with linear triangles. 1 Deformation Gradient rror Uniformly stabilized Adaptive ε=0.0 Adaptive ε= Fig. 3. (Left) Highly refined DG approximation using adaptive stabilization on 125 processors with trilinear hexahedra. The material is a compressible neo-hookean. The DG approximation has h = 1/36 resulting in 1.1(10 6 ) dof. Colored regions show where A is indefinite and thus where discontinuities are penalized. (Right) Comparison of the deformation gradient errors for adaptive stabilization and traditional stabilization with uniform penalty. The deformation gradient error is measured as F h F exact 0,B0 where F exact is the exact deformation gradient, and F h is its numerical approximation. For DG, the deformation gradient approximation is F h = D DG ϕ h. The mesh refinement is fixed with h = 1/12, and the errors are scaled by the maximum error observed. For each method, a decrease in β beyond what is shown resulted in a loss of stability. As β increases the uniform method quickly becomes sensitive whereas the adaptive method with ɛ = 0 is virtually insensitive and with ɛ = 0.01 a moderate reduction in error is seen for small values of β Although, traditional stabilization may perform well, several authors have shown that many DG methods are rather sensitive to the penalization parameter, β which is user defined. The methods displayed large variations in the accuracy of the discrete approximation to both ϕ and F when β is changed by very little. In an attempt to automatically determine β and optimize the accuracy of the approximation for a given h, we propose a method of adaptive stabilization. In this method, we take advantage of the indefiniteness β 4

5 of A. For a point X B 0 we set λ X = λ(a(f(x))), with λ(a) = max { } g : A : g ɛ, min 0 g R d d g : g (6) where ɛ R + and ɛ << 1 so that if A(F(X)) is positive definite then λ X is nearly zero, and if A(F(X)) has one or more negative eigenvalues then λ X is nearly equal to the absolute value of the smallest eigenvalue of A(F(X)). The newly stabilized method is formulated in terms of an incremental variational principle. At an increment k + 1 we add a stabilization term of the form β 2 T h λ k X R([ϕ h ]) 2 dv (7) where β > 0 is the stabilization parameter and λ k X is computed from the deformation obtained at increment k. Several authors have proved the equivalence of this term to the one shown in q. 5. In this case, the method is guaranteed to be stable provided that β is large enough. However, we have found for this method a suitable value of β is generally O(1). 0.1 DG Linear conforming 1 DG Linear conforming Displacement rror Deformation Gradient rror 0.1 1e Total DOF Total DOF Fig. 4. Convergence plot for the displacements (left) and deformation gradient (right) as a function of the number of degrees of freedom. The reference configuration is a neo-hookean square with unit length. Here we use ɛ = 0 and β = 1 with linear triangles. The displacement error is ϕ h ϕ exact 0,B0, the deformation gradient error is F h F exact 0,B0 where ϕ exact and F exact are calculated from a highly-refined conforming solution. In this case, the linear conforming approximation is more efficient than the proposed method for the computation of displacements, while the situation is reversed in the case of deformation gradient. In Fig. 3 the accuracy in the approximation to the deformation gradient is compared for the adaptive DG method and one using a traditional stabilization mechanism of the form β B 0 R([ϕ h ]) 2 dv. The standard stabilization uniformly penalizes discontinuities on each of the faces in the mesh, while the adaptive stabilization adds additional penalization to those areas in B 0 where A is indefinite. In this study we evaluate the effect of β on the accuracy of the DG approximation. Here reference configuration is a neo-hookean cube with unit length. A highly refined DG approximation to the deformation is shown in Fig. 3. For this particular deformation, simulations were done in parallel. To solve the linear system of equations we used Additive Schwarz preconditioning. For the preconditioner, an overlap of exactly one element (h) was implemented, and a direct LU factorization was used to solve the local problem. The linear solver and preconditioner were implemented within the PTSc framework. In the figure we clearly see the robustness and improved accuracy of the adaptive method for both ɛ = 0 and ɛ = In Fig. 4 we compare the accuracy as a function of the number of degrees of freedom in the mesh for the deformation gradient and the displacements. A DG and conforming approximation to the final deformation is shown in Fig.5. We measure the computational cost as the total number of degrees of freedom in the discrete problem. In the figure we see that the DG approximation to the deformation gradient is competitive with the conforming one. To more clearly see the advantage of using Wh d d to approximate derivatives, we 5

6 have plotted contours of D DG ϕ h for one DG approximation and ϕ h for a conforming approximation with the same number of degrees of freedom. In Fig. 5 we see that Wh d d yields piecewise linear interpolation while Vh d yields piecewise constant interpolation. Fig. 5. A coarse DG approximation (h = 1/9, left) and a refined conforming approximation (h = 1/24, right). The contours show the F 11 component of the approximation to the deformation gradient. Notice that the DG deformation gradient are piecewise linear inside each element, in contrast to the piecewise constant approximations in the conforming case. Both solutions have nearly the same error in deformation gradient computed with nearly the same number of degrees of freedom, DG has 972 and conforming In Fig. 6 we use DG with adaptive stabilization to model the elastin structure within a small cubic sample of arterial tissue with length 25(10 6 )m. The elastin structure was constructed from images obtained by Mary K. O Connell and Charles A. Taylor in the Cardiovascular Biomechanics Research Laboratory at Stanford. Due to the accuracy of DG and its non-locking behavior it is an ideal candidate for investigating arterial micromechanics. A highly refined DG mesh with 4.4(10) 6 dof on 343 processors is shown in Fig. 6. Fig. 6. (Left) lastin (red) suspended in soft extracellular matrix (blue). Both the elastin and extracellular matrix are modeled as a nearly incompressible linear elastic solid. (Right) Contours of the shear stress are shown when the sample is subjected to a transverse displacement of the upper surface. We see the DG approximation is able to capture some of the localizations that occur due to the rapidly oscillating material coefficients. The ability to capture stress localizations will be key when investigating phenomena such as aneurysm. 6