An Abstract Stabilization Method with Applications to Nonlinear Incompressible Elasticity

1 Introduction

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 [11]. Although all three papers discuss compressible and nearly incompressible nonlinear elasticity problems, they do not address the exactly incompressible problems.

In [12], 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 [13] 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 [14] 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 [5], 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 $Ω \subset R^{d} (1 \leq d \leq 3)$ for a d-dimensional bounded material body $B$ , the deformation of $B$ can be described by the map $^φ : Ω \to R^{d}$ defined by

^φ (X) = X +^u (X),

where $X = (X_{1}, \dots, X_{d})$ denotes the coordinates of a material point in the reference configuration and $^u (X)$

represents the corresponding displacement vector. Following the standard notation (see

[12]), we introduce the deformation gradient

^F

and the right Cauchy–Green deformation tensor

^C

by setting

^F = I + \nabla^u,^C = {^F}^{T}^F,

(2.1)

where I is the second-order identity tensor and $\nabla$ is the gradient operator with respect to the coordinate $X$ .

For a homogeneous neo-Hookean material, for example, latex and rubber, we define (see [15]) the potential energy function as

ψ (^u) = \frac{1}{2} μ [I :^C - d] - μ ln^J + \frac{λ}{2} (ln^J)^{2},

(2.2)

where $λ$ and $μ$ are positive constants, “:”represents the usual inner product for second-order tensors, and $^J = det^F$ is the deformation gradient Jacobian.

When we introduce he pressure-like variable (or simply pressure) $^p = λ ln^J$ , the potential energy (2.2) can be equivalently written as the following function of $^u$ and $^p$ (still denoted as $ψ$ with a little abuse of notation):

ψ (^u,^p) = \frac{1}{2} μ [I :^C - d] - μ ln^J +^p ln^J - \frac{1}{2 λ} (^p)^{2} .

When the body $B$ is subject to a given load $b = b (X)$ per unit volume in the reference configuration, the total elastic energy functional reads as

Π (^u,^p) = \int_{Ω} ψ (^u,^p) - \int_{Ω} b \cdot^u .

(2.3)

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 $^V$ and $^Q$ . The corresponding Euler–Lagrange equations arising from (2.3) lead to this solution: Find $(^u,^p) \in^V \times^Q$ such that

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} μ \int_{Ω}^F : \nabla v + \int_{Ω} (^p - μ) {^F}^{- T} : \nabla v = \int_{Ω} b \cdot v \forall v \in V, \int_{Ω} (ln^J - \frac{^p}{λ}) q = 0 \forall q \in Q, \end{matrix}

(2.4)

where $V$ and $Q$ 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

D J (^u) [v] = J (^u) F (^u)^{- T} : \nabla v = J (^u) F (^u) : \nabla v \forall v \in V .

We now focus on the case of an incompressible material, which corresponds to taking the limit $λ \to + \infty$ in (2.4). Hence, our problem becomes: Find $(^u,^p) \in^V \times^Q$ such that

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} μ \int_{Ω}^F : \nabla v + \int_{Ω} (^p - μ) {^F}^{- T} : \nabla v = \int_{Ω} b \cdot v \forall v \in V, \int_{Ω} q ln^J = 0 \forall q \in Q, \end{matrix}

(2.5)

or, in residual form: Find $(^u,^p) \in^V \times^Q$ such that

{Ru((^u,^p),v)=0  ∀v∈V,Rp((^u,^p),q)=0  ∀q∈Q,

(2.6)

where

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} R_{u} ((^u,^p), v) := μ \int_{Ω}^F : \nabla v + \int_{Ω} (^p - μ) {^F}^{- T} : \nabla v - \int_{Ω} b \cdot v, R_{p} ((^u,^p), q) := \int_{Ω} q ln^J . \end{matrix}

(2.7)

We now derive the linearization of problem (2.5) around a generic point $(^u,^p)$ . Observing that

D^F (^u) [u] = - {^F}^{- T} (\nabla u)^{T} {^F}^{- T} \forall u \in V,

we easily get the problem for the infinitesimal increment $(u, p)$ : Find $(u, p) \in V \times Q$ such that

(2.8)

where

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} ~ a (u, v) = μ \int_{Ω} \nabla u : \nabla v + \int_{Ω} (μ -^p) ({^F}^{- 1} \nabla u)^{T} : {^F}^{- 1} \nabla v, ~ b (u, q) = \int_{Ω} q {^F}^{- T} : \nabla u . \end{matrix}

(2.9)

Remark 2.1

Since problem (2.9) is the linearization of problem (2.5), it can be interpreted as the generic step of a Newton-like iteration procedure for the solution of the nonlinear problem (2.5).

Remark 2.2

Taking $(^u,^p) = (0, 0)$ in (2.9), we immediately recover the classical linear incompressible elasticity problem for small deformations; i.e., we find $(u, p) \in V \times Q$ such that

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 μ \int_{Ω} ϵ (u) : ϵ (v) + \int_{Ω} p div v = \int_{Ω} b \cdot v \forall v \in V, \int_{Ω} q div u = 0 \forall q \in Q, \end{matrix}

(2.10)

where $ϵ (u) = \frac{(\nabla u)^{T} + \nabla u}{2}$ denotes the symmetric gradient operator.

We now note that the Piola identity $div (^J {^F}^{- T}) = 0$ and $^J = 1$ give $div {^F}^{- T} = 0$ . Hence, we have

div ({^F}^{- T} u) = div {^F}^{- T} \cdot u + {^F}^{- T} : \nabla u = {^F}^{- T} : \nabla u .

Let ${^F}^{- T} u = w$ , we then consider the following problem: Find $(w, p) \in V \times Q$ such that

{a(w,v)+b(v,p)=−Ru((^u,^p),v) ∀v∈V,b(w,q)=−Rp((^u,^p),q)  ∀q∈Q,

(2.11)

where

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} a (w, v) = ~ a (^F w,^F v) = μ \int_{Ω} \nabla (^F w) : \nabla (^F v) + \int_{Ω} (μ -^p) ({^F}^{- 1} \nabla (^F w))^{T} : {^F}^{- 1} \nabla (^F v), b (w, p) = \int_{Ω} p div w . \end{matrix}

(2.12)

Let $V_{h} \subset V and Q_{h} \subset Q$ be the compatible finite element spaces. The discrete problem of (2.11) reads: Find $(w_{h}, p_{h}) \in V_{h} \times Q_{h}$ such that

{a(wh,vh)+b(vh,ph)=−Ru((^u,^p),vh) ∀vh∈Vh,b(wh,qh)=−Rp((^u,^p),qh)  ∀qh∈Qh,

(2.13)

where $a (\cdot, \cdot) and b (\cdot, \cdot)$ are defined by (2.12).

In some cases, it is possible that the continuous problem (2.11) is stable (or well-posed) but that the discrete problem (2.13) is not (see [5]). In such cases, we say that there is an unphysical instability caused by the discretization.

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 $V$ and $Q$ each be a Hilbert space, and assume that

a : V \times V \to R, b : V \times Q \to R

are continuous bilinear forms. Let $f \in V^{^{'}}$ and $g \in Q^{^{'}}$ . We denote both the dual pairing of $V$ with $V^{^{'}}$ and that of $Q$ with $Q^{^{'}}$ by $⟨ \cdot, \cdot ⟩$ . We consider the following problem: Find $(w, p) \in V \times Q$ such that

{\begin{matrix} a (w, v) + b (v, p) = ⟨ f, v ⟩ \forall v \in V, b (w, q) = ⟨ g, q ⟩ \forall q \in Q . \end{matrix}

(3.1)

We associate a mapping $B$ with the form $b$ :

B : V \to Q^{^{'}} : ⟨ B w, q ⟩ = b (w, q) \forall q \in Q .

Define

K = ker (B) := {v \in V : b (v, q) = 0 \forall q \in Q} .

As is well-known, the continuous problem (3.1) is well-posed under the following assumptions:

$A_{1} :$ the inf-sup condition, i.e., it holds that

$inf q \in Q sup v \in V \frac{b (v, q)}{∥ v ∥_{V} ∥ q ∥_{Q}} \geq β > 0,$ (3.2)

where $∥ \cdot ∥_{V} and ∥ \cdot ∥_{Q}$ are norms on the spaces $V$ and $Q$ , respectively;
$A_{2} :$ the coercivity on the kernel space, i.e., it holds that

$a (v, v) \geq α ∥ v ∥_{V}^{2} \forall v \in K, for some α > 0.$ (3.3)

Now we rewrite the problem (3.1) in an equivalent form as follows: Find $(w, p) \in V \times Q$ such that

{\begin{matrix} A (w, v) + b (v, p) = ⟨ f, v ⟩ + M (g, B v)_{Q^{^{'}}} \forall v \in V, b (w, q) = ⟨ g, q ⟩ \forall q \in Q, \end{matrix}

(3.4)

where $A (w, v) = a (w, v) + M (B w, B v)_{Q^{^{'}}}, (\cdot, \cdot)_{Q^{^{'}}}$ denotes the inner product on $Q^{^{'}}$ , and $M$ is a parameter to be chosen properly.

Let $V_{h} \subset V$ and $Q_{h} \subset Q$ be finite dimensional subspaces of $V$ and $Q$ such that the following assumption is satisfied:

$A_{3} :$ discrete inf-sup condition holds that

$inf q_{h} \in Q_{h} sup v_{h} \in V_{h} \frac{b (v_{h}, q_{h})}{∥ v_{h} ∥_{V} ∥ q_{h} ∥_{Q}} \geq β_{1} > 0,$ (3.5)

.

where $β_{1}$ is independent of $h$ .

We consider the discretization of the problem (3.4) as follows: Find $(w_{h}, p_{h}) \in V_{h} \times Q_{h}$ such that

{\begin{matrix} A (w_{h}, v_{h}) + b (v_{h}, p_{h}) = ⟨ f, v_{h} ⟩ + M (g, B v_{h})_{Q^{^{'}}} \forall v_{h} \in V_{h}, b (w_{h}, q_{h}) = ⟨ g, q_{h} ⟩ \forall q_{h} \in Q_{h} . \end{matrix}

(3.6)

Remark 3.1

Note that when $M = 0$ , (3.6) becomes the classical mixed finite element method. We call (3.6) the modified mixed finite element.

Lemma 3.1

(see [16]) $A_{1}$ is equivalent to the following condition, the operator $B : V^{⊥} \to Q^{^{'}}$ is an isomorphism, and

∥ B v ∥_{Q^{^{'}}} \geq β ∥ v ∥_{V} \forall v \in V^{⊥} .

(3.7)

Theorem 3.1

Under the assumptions $A_{1}$ and $A_{2}$ , there is a constant $M_{0}$ such that the discrete problem (3.6) is stable for any $M \geq M_{0}$ in the sense that there exists a positive constant $α_{1}$ such that

A (v_{h}, v_{h}) \geq α_{1} ∥ v_{h} ∥_{V}^{2} \forall v_{h} \in V_{h} .

(3.8)

For any $v_{h} \in V_{h}$ , noting that $V_{h} \subset V$ , we have $v_{h} = φ + φ^{⊥}$ , where $φ \in K, φ^{⊥} \in K^{⊥}$ and $K^{⊥}$ denotes the orthogonal of $K$ in $V$ . By Lemma 3.1, we have

$A (v_{h}, v_{h})$	$=$	$a (v_{h}, v_{h}) + M (B v_{h}, B v_{h})_{Q^{^{'}}}$
	$=$	$a (φ + φ^{⊥}, φ + φ^{⊥}) + M (B φ^{⊥}, B φ^{⊥})_{Q^{^{'}}}$
	$\geq$	$a (φ, φ) + a (φ, φ^{⊥}) + a (φ^{⊥}, φ) + a (φ^{⊥}, φ^{⊥}) + M β^{2} ∥ φ^{⊥} ∥_{V} .$

By assumption $A_{2}$ , the continuity of $a (\cdot, \cdot)$ and the Cauchy inequality, we obtain that

$A (v_{h}, v_{h})$	$\geq$	$α ∥ φ ∥_{V}^{2} - C_{1} ∥ φ ∥_{V} ∥ φ^{⊥} ∥_{V} - C_{2} ∥ φ^{⊥} ∥_{V}^{2} + M β^{2} ∥ φ^{⊥} ∥_{V}^{2}$
	$\geq$	$(α - ε C_{1}) ∥ φ ∥_{V}^{2} - (C_{2} + \frac{C_{1}}{} ε) ∥ φ^{⊥} ∥_{V}^{2} + M β^{2} ∥ φ^{⊥} ∥_{V}^{2}$
	$\geq$	$(α - ε C_{1}) ∥ φ ∥_{V}^{2} + (M β^{2} - (C_{2} + \frac{C_{1}}{ε})) ∥ φ^{⊥} ∥_{V}^{2} .$

Noting that $∥ v_{h} ∥_{V}^{2} = ∥ φ ∥_{V}^{2} + ∥ φ^{⊥} ∥_{V}^{2}$ and choosing $ε = \frac{α}{2 C_{1}}, M_{0} = \frac{\frac{α}{2} + C_{2} + \frac{2 C_{1}^{2}}{α}}{β^{2}}$ , we have

A (v_{h}, v_{h}) \geq α_{1} (∥ φ ∥_{V}^{2} + ∥ φ^{⊥} ∥_{V}^{2}) = α_{1} ∥ v_{h} ∥_{V}^{2},

which completes the proof.

Furthermore, noting that $A (\cdot, \cdot)$ is continuous, by the classic Brezzi theory for the saddle point problem [16, 17, 18], we have

Theorem 3.2

Let $w, p$ be the solution of the problem (3.4), then under the assumptions $A_{1}, A_{2}, A_{3}$ and providing that $M$ is sufficiently large, the discrete problem (3.6) has a unique solution $(w_{h}, p_{h}) \in V_{h} \times Q_{h}$ , which satisfies

∥ w - w_{h} ∥_{V} + ∥ p - p_{h} ∥_{Q} \leq C (inf v_{h} \in V_{h} ∥ w - v_{h} ∥_{V} + inf q_{h} \in Q_{h} ∥ p - q_{h} ∥_{Q}),

(3.9)

where $C$ is a constant depending on $α_{1}$ and $β_{1}$ .

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 $Ω$ is connected, we consider the d-dimensional Dirichlet boundary value problem (other boundary value problems are similar) of (2.11), which means that $V = (H_{0}^{1} (Ω))^{d}, Q = L_{0}^{2} (Ω)$ . We choose $V_{h} and Q_{h}$ as finite element spaces such that $A_{3}$ is satisfied. Furthermore, $K = {v \in (H_{0}^{1} (Ω))^{d} : div v = 0}$ . We rewrite (2.11) as: Find $(w, p) \in V \times Q$ such that

{A(w,v)+b(v,p)=−Ru((^u,^p),v)−MRp((^u,^p),divv)  ∀v∈V,b(w,q)=−Rp((^u,^p),q)  ∀q∈Q,

(4.1)

where $A (w, v) = a (w, v) + M (div w, div v)$ , $a (\cdot, \cdot) and b (\cdot, \cdot)$ are defined by (2.12), $(\cdot, \cdot)$ denotes the inner product on $L^{2} (Ω)$ , and $M$ is a parameter chosen properly.

The discretization of the problem (4.1) is as follows: Find $(w_{h}, p_{h}) \in V_{h} \times Q_{h}$ such that

{A(wh,vh)+b(vh,ph)=−Ru((^u,^p),vh)−MRp((^u,^p),divvh)  ∀vh∈Vh,b(wh,qh)=−Rp((^u,^p),qh)  ∀qh∈Qh.

(4.2)

Lemma 4.1

([19], Corollary 2.3) We have

K^{⊥} = {(- △)^{- 1} grad f : f \in L_{0}^{2} (Ω)} .

Let $K^{0} = {y \in (H^{- 1} (Ω))^{d} :< y, ϕ >= 0 \forall ϕ \in K}$ , where $< \cdot, \cdot >$ denotes the duality paring between $(H^{- 1} (Ω))^{d}$ and $(H_{0}^{1} (Ω))^{d}$ .

Lemma 4.2

([19], Corollary 2.4) Let $Ω$ be connected. Then

the operator grad is an isomorphism from $L_{0}^{2} (Ω)$ onto $K^{0}$ , and
the operator div is an isomorphism from $K^{⊥}$ onto $L_{0}^{2} (Ω)$ .

Theorem 4.1

If $a (\cdot, \cdot)$ defined by (2.12) satisfies

a (v, v) \geq α ∥ v ∥_{1}^{2} \forall v \in K, for some α > 0,

the discrete problem (4.2) is stable in the sense that there exists a positive constant $α_{1}$ such that

A (v_{h}, v_{h}) \geq α_{1} ∥ v_{h} ∥_{1}^{2} \forall v_{h} \in V_{h} .

(4.3)

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).

Theorem 4.2

Let $w, p$ be the solution of the problem (2.11), then under the assumptions $A_{2}$ and $A_{3}$ and providing that $M$ is sufficiently large, the discrete problem (4.2) has a unique solution $(w_{h}, p_{h}) \in V_{h} \times Q_{h}$ , which satisfies

∥ w - w_{h} ∥_{1} + ∥ p - p_{h} ∥_{0} \leq C (inf v_{h} \in V_{h} ∥ w - v_{h} ∥_{1} + inf q_{h} \in Q_{h} ∥ p - q_{h} ∥_{0}),

(4.4)

where the constant $C$ depends on $α_{1}, β_{1}$ .

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 $(X, Y)$ , we consider a square material body whose reference configuration is $Ω = (- 1, 1) \times (- 1, 1)$ . We denote $Γ = [- 1, 1] \times {1}$ as the upper part of its boundary, while the remaining part of $\partial Ω$ is denoted with $Γ_{D}$ . The total energy is assumed to be as in (2.3), where the external loads are given by the vertical uniform body forces: $b = γ f$ , where $f = (0, 1)^{⊤}$ . The two problems differ in regard to the imposed boundary conditions. More precisely:

Problem 1. We set clamped boundary conditions on $Γ_{D}$ , but traction-free boundary conditions on $Γ$ .
Problem 2. We set vanishing normal displacements on $Γ_{D}$ , but traction-free boundary conditions on $Γ$ .

It is easy to see that both problems admit a trivial solution for every $γ \in R, i . e . (^u,^p) = (0, γ r)$ , where $r = r (X, Y) = 1 - Y$ .

For the problems under investigation, the corresponding linearized problems (cf. (2.12)) can both be written as: Find $(w, p) \in V \times Q$ such that

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} 2 μ \int_{Ω} ϵ (w) : ϵ (v) - γ \int_{Ω} r (\nabla w)^{⊤} : \nabla (v) + \int_{Ω} p div v = δ γ \int_{Ω} f \cdot v, \forall v \in V, \int_{Ω} q div w = 0 \forall q \in Q . \end{matrix}

(5.1)

where $δ γ$ is the increment of the parameter of $γ$ .

For these two different problems, the spaces $V$ and $Q$ are defined as follows:

Problem 1. $V = {v \in H^{1} (Ω)^{2} : v |_{Γ_{D}} = 0}; Q = L^{2} (Ω)$ .
Problem 2. $V = {v \in H^{1} (Ω)^{2} : (v \cdot n) |_{Γ_{D}} = 0}; Q = L^{2} (Ω)$ , where $n$ denotes the outward normal vector.

The stable discrete formulation reads as follows (see (4.2)): Find $(w_{h}, p_{h}) \in V_{h} \times Q_{h}$ such that

{\begin{matrix} A (w_{h}, v_{h}) + b (v_{h}, p_{h}) = δ γ \int_{Ω} f \cdot v_{h}, \forall v_{h} \in V_{h}, b (w_{h}, q_{h}) = 0, \forall q_{h} \in Q_{h} . \end{matrix}

(5.2)

where $A (w, v) = 2 μ \int_{Ω} ϵ (w) : ϵ (v) - γ \int_{Ω} r (\nabla w)^{⊤} : \nabla (v) + M (γ) \int_{Ω} div w div v$ ;
$b (w, p) = \int_{Ω} p div v$ . Here $M (γ)$ is a parameter chosen depending on $γ$ .

Remark 5.1

For Problem $1$ , it has been theoretically proved in [5] that the continuous problem (5.1) is stable when $γ < 3 μ$ .

5.2 The MINI element

The considered scheme for (5.2) is the MINI element (see [18]). Let $T_{h}$ be a triangular mesh of $Ω$ with the mesh size $h$ . For the discretization of the displacement field, we take

V_{h} = {v_{h} \in V : v_{h} |_{T} \in P_{1} (T)^{2} + B (T)^{2} \forall T \in T_{h}},

where $P_{1} (T)$ is the space of linear functions on $T$ , and $B (T)$ is the linear space generated by $b_{T}$ , the standard cubic bubble function on $T$ . For the pressure discretization, we take

Q_{h} = {q_{h} \in H^{1} (Ω) \cap Q : q_{h} |_{T} \in P_{1} (T) \forall T \in T_{h}} .

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 [5] that for Problem $1$ the classical mixed finite element method is stable when $γ < μ$ , but unstable when $γ > \frac{3}{2} μ$ . In this numerical experiment, we will demonstrate that the modified mixed finite element method is stable for both Problem $1$ and Problem $2$ when $- \infty < γ < 3 μ$ (which is the stability range for the continuous case of Problem $1$ in [5]) as predicated by our Theorem 3.1.

Noting Theorem 3.1

, we study the eigenvalues of the matrix induced by the bilinear form

A (\cdot, \cdot)

for the problems under consideration. The first loads for which we find a negative eigenvalue are the critical ones. We start from

γ = 0

for both positive and negative loading conditions, i.e., for

γ < 0

and

γ > 0

. We indicate as the critical loads

γ_{m, h}

and

γ_{M, h}

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

{~ γ}_{m, h}

and

{~ γ}_{M, h}

, respectively, where

~ γ = \frac{γ L}{μ}

. Here,

L

is some problem characteristic length, set equal to

1

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

(~ γ > 10^{6})

, we set

~ γ = \infty

. Noting that the bound constant

C_{1}

for the formulation

a (\cdot, \cdot)

is of order

γ + 2 μ

and the stable constant of

a (\cdot, \cdot)

is of order

μ

, we can choose

M_{0} = m_{1} | ~ γ | + m_{2} {~ γ}^{2}

by the proof of Theorem 3.1. Hence, in the codes, we set

μ = 40, M (γ) = m_{1} | ~ γ | + m_{2} {~ γ}^{2}

, then we adjust the parameter

~ γ

m_{1}

and

m_{2}

to investigate the stability performance. Furthermore, we take

m_{1} = 320, m_{2} = 0

for Problem

1

and

m_{1} = 320, m_{2} = 1.36

for Problem

2

Nodes	${~ γ}_{m, h}$	${~ γ}_{M, h}$
$5 \times 5$	$- \infty$	$+ \infty$
$9 \times 9$	$- \infty$	$14.50$
$17 \times 17$	$- \infty$	$8.25$
$33 \times 33$	$- \infty$	$7.13$

Table 1: Stability Limits for Problem

1

Nodes	${~ γ}_{m, h}$	${~ γ}_{M, h}$
$5 \times 5$	$- \infty$	$+ \infty$
$9 \times 9$	$- \infty$	$3.88$
$17 \times 17$	$- \infty$	$3.38$
$33 \times 33$	$- \infty$	$3.23$

Table 2: Stability Limits for Problem

2

To verify the convergence result (3.2), we set another data for (5.1) that $f = (- e^{x} (1 - y), e^{x})^{⊤}$ and the true solution is $(w, p) = (0, δ γ e^{x} (1 - y))$ .

Nodes	$∥ p - p_{h} ∥_{0}$	$∥ w - w_{h} ∥_{1}$	$o r d e r$
$5 \times 5$	$6.0469 \times 10^{- 2}$	$1.0518 \times 10^{- 6}$	$- -$
$9 \times 9$	$1.5141 \times 10^{- 2}$	$1.8890 \times 10^{- 7}$	$2$
$17 \times 17$	$3.7871 \times 10^{- 3}$	$3.0897 \times 10^{- 8}$	$2$
$33 \times 33$	$9.4692 \times 10^{- 4}$	$9.2283 \times 10^{- 9}$	$2$

Table 3: The Convergence of

~ γ = 7.125

for Problem

1

Nodes	$∥ p - p_{h} ∥_{0}$	$∥ w - w_{h} ∥_{1}$	$o r d e r$
$5 \times 5$	$6.0187 \times 10^{- 2}$	$7.5563 \times 10^{- 6}$	$- -$
$9 \times 9$	$1.5129 \times 10^{- 2}$	$1.6314 \times 10^{- 6}$	$2$
$17 \times 17$	$3.7867 \times 10^{- 3}$	$3.3616 \times 10^{- 7}$	$2$
$33 \times 33$	$9.4691 \times 10^{- 4}$	$7.7049 \times 10^{- 8}$	$2$

Table 4: The convergence of

~ γ = 3.23

for Problem

2

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 ${~ γ}_{M, h}$ are competitive to the “exact” values claimed in [12]. 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 $1$ when $~ γ = 7.125$ , and hence the convergence fails. However, the modified method performs well.

6 Conclusions

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 $M$ , 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 $M_{0}$ 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 Section

References

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An Abstract Stabilization Method with Applications to Nonlinear Incompressible Elasticity

Authors

Fully discrete finite element approximation for a family of degenerate parabolic mixed equations

An extended mixed finite element method for elliptic interface problems

Conforming and Nonconforming Finite Element Methods for Biharmonic Inverse Source Problem

Functional sets with typed symbols: Framework and mixed Polynotopes for hybrid nonlinear reachability and filtering

Mixed finite element approximation of periodic Hamilton–Jacobi–Bellman problems with application to numerical homogenization

Analysis of an abstract mixed formulation for viscoelastic problems

A mixed method for 3D nonlinear elasticity using finite element exterior calculus

1 Introduction

2 Motivation: The finite strain incompressible elasticity problem

Remark 2.1

Remark 2.2

3 Abstract stabilization strategy for mixed approximation

Remark 3.1

Lemma 3.1

Theorem 3.1

Theorem 3.2

4 Application to the nonlinear elasticity problem

Lemma 4.1

Lemma 4.2

Theorem 4.1

Theorem 4.2

5 Numerical experiment

5.1 Two examples

Remark 5.1

5.2 The MINI element

5.3 Numerical results

6 Conclusions

References

An Abstract Stabilization Method with Applications to Nonlinear Incompressible Elasticity

Authors

Related Research

Fully discrete finite element approximation for a family of degenerate parabolic mixed equations

An extended mixed finite element method for elliptic interface problems

Conforming and Nonconforming Finite Element Methods for Biharmonic Inverse Source Problem

Functional sets with typed symbols: Framework and mixed Polynotopes for hybrid nonlinear reachability and filtering

Mixed finite element approximation of periodic Hamilton–Jacobi–Bellman problems with application to numerical homogenization

Analysis of an abstract mixed formulation for viscoelastic problems

A mixed method for 3D nonlinear elasticity using finite element exterior calculus

This week in AI

1 Introduction

2 Motivation: The finite strain incompressible elasticity problem

Remark 2.1

Remark 2.2

3 Abstract stabilization strategy for mixed approximation

Remark 3.1

Lemma 3.1

Theorem 3.1

Theorem 3.2

4 Application to the nonlinear elasticity problem

Lemma 4.1

Lemma 4.2

Theorem 4.1

Theorem 4.2

5 Numerical experiment

5.1 Two examples

Remark 5.1

5.2 The MINI element

5.3 Numerical results

6 Conclusions

References