Higher order stabilized finite element method for hyperelastic finite deformation

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Abstract

This paper presents a higher order stabilized finite element formulation for hyperelastic large deformation problems involving incompressible or nearly incompressible materials. A Lagrangian finite element formulation is presented where mesh dependent terms are added element-wise to enhance the stability of the mixed finite element formulation. A reconstruction method based on local projections is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor. Linearization of the weak form is derived to enable a Newton–Raphson solution procedure of the resulting non-linear equations. Numerical experiments using the stabilization method with equal order shape functions for the displacement and pressure fields in hyperelastic problems show that the stabilized method is effective for some non-linear finite deformation problems. Finally, conclusions are inferred and extensions of this work are discussed.

Introduction

Stabilized finite element methods (SFEMs) consist of adding mesh dependent terms to the usual Galerkin method (see, for example, the works of [1], [2]) in order to improve the stability of the solution in problems with known instabilities, for example in modeling incompressible material deformation. Without the added stabilization terms, Galerkin methods applied to incompressible or nearly incompressible material behavior in the setting of a mixed finite element method must fulfill the Ladyzenskaya–Babuska–Brezzi (LBB) condition to achieve unique solvability, convergence and robustness [3]. This places severe restrictions on the choice of the solution space. Without balancing the interpolation functions according to the LBB condition, large errors or oscillations may appear in the solution. Furthermore, the LBB condition complicates the use of p-adaptive techniques, where flexibility is needed in varying the interpolation order. The advantage of stabilized methods is that they do not impose constraints on the interpolation functions, thus allowing for greater flexibility in balancing efficiency and accuracy.

SFEMs, which avoid the limitations on the interpolation functions in the traditional Galerkin finite element method, are widely used now, especially in applications in fluid mechanics. The stabilization terms, which are added to the usual Galerkin formulation, are functions of the residuals of the Euler–Lagrange equations evaluated element-wise. From the construction, it follows that consistency is not affected since the exact solution satisfies both the Galerkin term and the additional terms (see, for example, the works of [1], [4], [5]). Recently, Klaas et al. [6] have applied a stabilized finite element method with linear elements to hyperelasticity, and the results show the effectiveness of the stabilized finite element method for large deformation problems. However, higher order stabilized finite elements were not implemented in that work because of the difficulty in computing the required higher order derivatives.

In this paper, a higher order stabilized finite element formulation in a Lagrangian reference frame for hyperelasticity is developed. A Petrov–Galerkin method, following that presented in [1], is used. This results in a term with the strong form of the equilibrium equation being added to the usual Galerkin method. This term involves the divergence of part of the stress tensor, which, in turn, depends on the deformation gradient field. In this work, a local reconstruction method, following the work presented for two-dimensional Navier–Stokes equations in [7], is used to compute the stress at the finite element nodal locations. Then the divergence of the reconstructed stress is computed from the nodal quantities and shape functions. Numerical examples with a non-linear hyperelastic constitutive law for rubber-like materials are given. Finally, conclusions are inferred and directions for future work are discussed.

Section snippets

Governing equations for Lagrangian description

Consider a three-dimensional reference domain B with boundary Γ. The boundary value problem for finite elasticity in the absence of body forces defined on the reference configuration is given as

Find a displacement field u such thatDiv(FS)=0inBwith boundary conditions(FS)n=gonΓN,u=ūonΓD,where F is the deformation gradient, S is the second Piola–Kirchoff stress, n is the outward unit normal and g is the prescribed traction load on ΓN, and ū is the prescribed displacement on ΓD. Finally, for

Stabilized mixed displacement–pressure formulation

Following standard Galerkin procedure, the strong form in Eq. (1) is integrated with a kinematically admissible weighting function wBDiv(FSwdV=0.In this work, a simple Petrov–Galerkin formulation following that presented in [6] is used. Specifically, following the standard Galerkin method, the strong form of the equilibrium equation in Eq. (1) is integrated with weight function w lying in the kinematically admissible space W=V×P:w=uFT∇p,where u is the standard part and the second

Linearization

Now Eq. (11), after the local reconstruction (Eq. (14)), can be rewritten in a short form asR(u,p)(u)=Lext(u),S(u,p)(p)=F(p).Linearization of Eq. (16), leads to the stabilized system equationsk(u,p)u,u)+g(u)(Δp,u)=Lext(u)−R(u,p)(u),h(u,p)u,p)+d(u)(Δp,p)=F(p)−S(u,p)(p),wherek(u,p)u,u)=∫B4FT∇(Δu)sym:2W̃(C)CC+p2J(u)CC:FTusymdV+∫B2W̃(C)C+pJ(u)C:[∇(Δu)Tu]dV,g(u)(Δp,u)=∫B2ΔpJ(u)C:FTudV,h(u,p)u,p)=∫B2pU″(J(u))J(u)C:FT∇(Δu)dV−∑nele=1αhe2B4[∇p⊗∇p]sym:2J(u)

Examples

The above stabilized formulation has been implemented into an object-oriented finite element framework, named Trellis, developed in the Scientific Computation Research Center at Rensselaer Polytechnic Institute, see [8]. In order to provide a quantitative assessment of the above stabilized, mixed finite element formulation and to demonstrate the behavior of the higher order stabilized mixed method, two examples are investigated: (1) Cook's example, and (2) plane strain extension of a

Conclusions

A higher order, stabilized, Lagrangian finite element formulation for nearly incompressible finite deformation hyperelasticity is presented. In the stabilized formulation, mesh dependent terms, which enhance stability, are added element-wise to the usual Galerkin method resulting in a Petrov–Galerkin formulation. A local reconstruction method is used to compute the higher order derivatives that arise in the stabilization terms, specifically derivatives of the stress tensor.

Numerical examples,

Acknowledgements

This work has been supported by the National Science Foundation through grant DMI-9634920 as well as by the US Department of Defense – Department of the Air Force, Wright Laboratory.

References (8)

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