Many finite element schemes perform very well in terms of both accuracy and stability for linear elastic problems (see [1, 2, 3, 4]), including in highly constrained situations such as incompressible cases. However, it is well-known that though such schemes can be extended to nonlinear elasticity cases (for instance, large deformation elasticity problems) the same level of stability is by no means guaranteed (see [5, 6, 7, 8]).
In [9, 10], a stabilized discontinuous Galerkin method for nonlinear compressible elasticity was introduced. Because the stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities, it is called an adaptive stabilization strategy. This same adaptive stabilization strategy can also be found in . Although all three papers discuss compressible and nearly incompressible nonlinear elasticity problems, they do not address the exactly incompressible problems.
In , an isogeometric “stream function” formulation was used to exactly enforce the linearized incompressibility constraint. In fact, the exact satisfaction of the linearized incompressibility constraint leads to a numerical approximation of the stability range that converges to the exact one (i.e., the one for the continuum problem). As the stream function is given by a fourth-order PDE, the high regularity of the NURBS shape functions must be used there and the stream function in 3dimensions is not unique.
Brezzi, Fortin, and Marini  presented a modified mixed formulation for second-order elliptic equations and linear elasticity problems that automatically satisfied the coercivity condition on the discrete level. Using the same technique and applying stable Stokes elements to a modified formulation for Darcy-Stokes-Brinkman models, Xie, Xu and Xue  obtained a uniformly stable method.
In this paper, motivated by the modified formulation used in [13, 14] for second-order equations and the Darcy-Stokes-Brinkman models respectively, we devise a stabilization strategy for the classical mixed finite method designed for Stokes equations and obtain a modified method for nonlinear incompressible elasticity. As indicated in , a usual mixed finite element method that is stable for Stokes equations such as the MINI element is not stable for nonlinear incompressible elasticity even though the continuous problem is stable. We note that, for the usual mixed finite method, the unphysical instability is caused by the fact that the discrete solution is not exactly divergence-free. Hence, we rewrite the continuous problem. Based on that we design a stabilization strategy involving the divergence of the discrete solution. We prove that the modified mixed finite element method can remove the unphysical instability and lead to optimal convergence. Hence, all stable Stokes elements can also be made stable for discrete nonlinear elasticity problems.
The rest of the paper is organized as follows. In section 2, we present the finite strain incompressible elasticity problem and obtain the linearized formulation of the nonlinear elasticity problem. In section 3, we propose the stabilization strategy and derive the modified mixed finite element method in an abstract framework, and in section 4 we apply this modified method to the nonlinear incompressible elasticity problem. Furthermore, we obtain an optimal approximation of the modified finite element method. In section 5, we give two simple examples and the corresponding numerical experiments to verify the stability and accuracy of the modified finite element method. We present our conclusions in section 6.
2 Motivation: The finite strain incompressible elasticity problem
To study the finite strain incompressible elasticity problem, we adopt what is known as the material description in this paper. Given a reference configuration for a d-dimensional bounded material body , the deformation of can be described by the map defined by
where denotes the coordinates of a material point in the reference configuration and
represents the corresponding displacement vector. Following the standard notation (see), we introduce the deformation gradient
and the right Cauchy–Green deformation tensorby setting
where I is the second-order identity tensor and is the gradient operator with respect to the coordinate .
For a homogeneous neo-Hookean material, for example, latex and rubber, we define (see ) the potential energy function as
where and are positive constants, “:”represents the usual inner product for second-order tensors, and is the deformation gradient Jacobian.
When we introduce he pressure-like variable (or simply pressure) , the potential energy (2.2) can be equivalently written as the following function of and (still denoted as with a little abuse of notation):
When the body is subject to a given load per unit volume in the reference configuration, the total elastic energy functional reads as
According to the standard variational principles, the equilibrium is derived by searching for critical points of (2.3) in suitable admissible displacement and pressure spaces and . The corresponding Euler–Lagrange equations arising from (2.3) lead to this solution: Find such that
where and are the admissible variation spaces for the displacements and pressures, respectively. Also note that in (2.4), the linearization of the deformation gradient Jacobian is
We now focus on the case of an incompressible material, which corresponds to taking the limit in (2.4). Hence, our problem becomes: Find such that
or, in residual form: Find such that
We now derive the linearization of problem (2.5) around a generic point . Observing that
we easily get the problem for the infinitesimal increment : Find such that
Taking in (2.9), we immediately recover the classical linear incompressible elasticity problem for small deformations; i.e., we find such that
where denotes the symmetric gradient operator.
We now note that the Piola identity and give . Hence, we have
Let , we then consider the following problem: Find such that
Let be the compatible finite element spaces. The discrete problem of (2.11) reads: Find such that
where are defined by (2.12).
3 Abstract stabilization strategy for mixed approximation
As shown at the end of the previous section, it is necessary to establish stable discretization when the continuous problem (2.11) is stable. So in this section, we propose an abstract stability strategy framework for the mixed approximation.
Let and each be a Hilbert space, and assume that
are continuous bilinear forms. Let and . We denote both the dual pairing of with and that of with by . We consider the following problem: Find such that
We associate a mapping with the form :
As is well-known, the continuous problem (3.1) is well-posed under the following assumptions:
the inf-sup condition, i.e., it holds that
where are norms on the spaces and , respectively;
the coercivity on the kernel space, i.e., it holds that
Now we rewrite the problem (3.1) in an equivalent form as follows: Find such that
where denotes the inner product on , and is a parameter to be chosen properly.
Let and be finite dimensional subspaces of and such that the following assumption is satisfied:
discrete inf-sup condition holds that
where is independent of .
We consider the discretization of the problem (3.4) as follows: Find such that
(see ) is equivalent to the following condition, the operator is an isomorphism, and
Under the assumptions and , there is a constant such that the discrete problem (3.6) is stable for any in the sense that there exists a positive constant such that
For any , noting that , we have , where and denotes the orthogonal of in . By Lemma 3.1, we have
By assumption , the continuity of and the Cauchy inequality, we obtain that
Noting that and choosing , we have
which completes the proof.
4 Application to the nonlinear elasticity problem
We now apply the abstract stabilized framework proposed in the previous section to nonlinear incompressible elasticity problems. Suppose that is connected, we consider the d-dimensional Dirichlet boundary value problem (other boundary value problems are similar) of (2.11), which means that . We choose as finite element spaces such that is satisfied. Furthermore, . We rewrite (2.11) as: Find such that
where , are defined by (2.12), denotes the inner product on , and is a parameter chosen properly.
The discretization of the problem (4.1) is as follows: Find such that
(, Corollary 2.3) We have
Let , where denotes the duality paring between and .
(, Corollary 2.4) Let be connected. Then
the operator grad is an isomorphism from onto , and
the operator div is an isomorphism from onto .
By the abstract framework, the proof is obvious.
Also from the abstract framework in the previous section, we can get an approximate accuracy result similar to (3.9).
5 Numerical experiment
In this section, we report our numerical experiments in regard to the performance of the stabilization strategy for two simple examples. We consider here simple problems using some mixed finite element formulations that are known to be optimal for Stokes equations. We first briefly present the finite element under consideration and then show the numerical results obtained using such element.
5.1 Two examples
In this subsection, we present two simple problems that will be used in subsection 5.3 to discuss the performance of the stabilization strategy proposed in Section 3. Using the usual Cartesian coordinates , we consider a square material body whose reference configuration is . We denote as the upper part of its boundary, while the remaining part of is denoted with . The total energy is assumed to be as in (2.3), where the external loads are given by the vertical uniform body forces: , where . The two problems differ in regard to the imposed boundary conditions. More precisely:
Problem 1. We set clamped boundary conditions on , but traction-free boundary conditions on .
Problem 2. We set vanishing normal displacements on , but traction-free boundary conditions on .
It is easy to see that both problems admit a trivial solution for every , where .
For the problems under investigation, the corresponding linearized problems (cf. (2.12)) can both be written as: Find such that
where is the increment of the parameter of .
For these two different problems, the spaces and are defined as follows:
Problem 1. .
Problem 2. , where denotes the outward normal vector.
The stable discrete formulation reads as follows (see (4.2)): Find such that
. Here is a parameter chosen depending on .
5.2 The MINI element
where is the space of linear functions on , and is the linear space generated by , the standard cubic bubble function on . For the pressure discretization, we take
5.3 Numerical results
We now study the stability performance of the discretized model problems by means of the modified mixed finite element formulations briefly described above. It has been theoretically proved and numerically verified in  that for Problem the classical mixed finite element method is stable when , but unstable when . In this numerical experiment, we will demonstrate that the modified mixed finite element method is stable for both Problem and Problem when (which is the stability range for the continuous case of Problem in ) as predicated by our Theorem 3.1.
Noting Theorem 3.1
, we study the eigenvalues of the matrix induced by the bilinear formfor the problems under consideration. The first loads for which we find a negative eigenvalue are the critical ones. We start from for both positive and negative loading conditions, i.e., for and . We indicate as the critical loads and the first load values for which we find a negative eigenvalue. A subsequent bisection-type procedure is used to increase the accuracy of the critical load detections. The corresponding nondimensional quantities are denoted with and , respectively, where . Here, is some problem characteristic length, set equal to for simplicity, consistents also with the geometry of the model problems. If we do not detect any negative eigenvalue for extremely large values of the load multiplier , we set . Noting that the bound constant for the formulation is of order and the stable constant of is of order , we can choose by the proof of Theorem 3.1. Hence, in the codes, we set , then we adjust the parameter , and to investigate the stability performance. Furthermore, we take for Problem and for Problem .
Table 1 and Table 2 show that the stabilization strategy together with the corresponding modified mixed finite element method proposed in this paper is effective. The stability performance can be improved obviously. In fact, these values of are competitive to the “exact” values claimed in . Furthermore, we can see that the modified mixed finite element method is also locking-free by verifying the convergence results (3.2), which are shown in Table 3 and Table 4. In fact, the classical mixed finite element method is unstable for the Problem when , and hence the convergence fails. However, the modified method performs well.
Within the framework of finite elasticity for incompressible materials, it is well known that the classical mixed finite element discretization can sometimes be unstable even though the continuous problem is stable. In this paper, we reformulated the continuous problem and proposed an abstract stabilization strategy based on the new continuous formulation and obtained a modified mixed finite element method. We proved theoretically in Section 3 that for a sufficiently large , the modified mixed finite element method is stable whenever the continuous problem is stable, and the method maintains the optimal convergence of the classical one. We verified by numerical experiments in Section 5 that the modified mixed finite element method is much more stable than the classical one and is also locking-free. The provided in Section 3
always overestimates the parameter used in practical problems. However, we can choose the parameter heuristically by analyzing the stability and continuity of continuous problems in the numerical experiments presented in Section5.
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