Pure Displacement Problem Research Articles

On Korn’s First Inequality in a Hardy-Sobolev Space

AbstractKorn’s first inequality states that there exists a constant such that the ${\mathcal {L}}^{2}$ L 2 -norm of the infinitesimal displacement gradient is bounded above by this constant times the ${\mathcal {L}}^{2}$ L 2 -norm of the infinitesimal strain, i.e., the symmetric part of the gradient, for all infinitesimal displacements that are equal to zero on the boundary of a body ℬ. This inequality is known to hold when the ${\mathcal {L}}^{2}$ L 2 -norm is replaced by the ${\mathcal {L}}^{p}$ L p -norm for any $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) . However, if $p=1$ p = 1 or $p=\infty $ p = ∞ the resulting inequality is false. It was previously shown that if one replaces the ${\mathcal {L}}^{\infty}$ L ∞ -norm by the $\operatorname{BMO}$ BMO -seminorm (Bounded Mean Oscillation) then one maintains Korn’s inequality. (Recall that ${\mathcal {L}}^{\infty}({\mathcal {B}})\subset \operatorname{BMO}({\mathcal {B}}) \subset {\mathcal {L}}^{p}({\mathcal {B}})\subset {\mathcal {L}}^{1}({ \mathcal {B}})$ L ∞ ( B ) ⊂ BMO ( B ) ⊂ L p ( B ) ⊂ L 1 ( B ) , $1< p<\infty $ 1 < p < ∞ .) In this manuscript it is shown that Korn’s inequality is also maintained if one replaces the ${\mathcal {L}}^{1}$ L 1 -norm by the norm in the Hardy space ${\mathcal {H}}^{1}$ H 1 , the predual of $\operatorname{BMO}$ BMO . One caveat: the results herein are only applicable to the pure-displacement problem with the displacement equal to zero on the entire boundary of ℬ.

Quasi-optimal convergence of AFEM based on separate marking, Part I

AbstractVarious applications in computational fluid dynamics and solid mechanics motivate the development of reliable and efficient adaptive algorithms for nonstandard finite element methods (FEMs). Standard adaptive finite element algorithms consist of the iterative loop of the basic steps Solve, Estimate, Mark and Refine. For separate marking strategies, this standard scheme may be universalised. The (total) error estimator is split into a volume term and an error estimator term.Since the volume term is independent of the discrete solution, an appropriate data approximation may be realised by a high degree of local mesh refinement. This observation results in a natural adaptive algorithm based on separate marking. Its quasi-optimal convergence is proven in this first part for the Poisson model problem and nonconforming Crouzeix-Raviart FEM. The numerical experiments confirm the optimal convergence rates and reveal its flexibility. The second part is devoted to the quasi-optimality of S-AFEM-AA and the pure displacement problem in linear elasticity and the Stokes equations.

Quasi-optimal convergence of AFEM based on separate marking, Part II

AbstractVarious applications in computational fluid dynamics and solid mechanics motivate the development of reliable and efficient adaptive algorithms for nonstandard finite element methods (FEMs). Standard adaptive finite element algorithms consist of the iterative loop of the basic steps Solve, Estimate, Mark, and Refine. For separate marking strategies, this standard scheme may be universalised. The (total) error estimator is split into a volume term and an error estimator term.Since the volume term is independent of the discrete solution, an appropriate data approximation may be realised by a high degree of local mesh refinement. This observation results in a natural adaptive algorithm based on separate marking. Its quasi-optimal convergence is proven in this second part for the pure displacement problem in linear elasticity and the Stokes equations and nonconforming Crouzeix-Raviart FEM. The proofs follow the same general methodology as for the Poisson model problem in the first part of this series. The numerical experiments confirm the optimal convergence rates and reveal its flexibility.

The Adaptive Nonconforming FEM for the Pure Displacement Problem in Linear Elasticity is Optimal and Robust

This paper presents a natural nonconforming adaptive finite element algorithm and proves its quasi-optimal complexity for the pure displacement Navier–Lamé equations. The convergence rates are robust with respect to the Lamé parameter $\lambda \to \infty$ in the sense that all constants in the quasi-optimal convergence rate stay bounded for almost incompressible materials and so the Stokes equations are covered by our analysis in the limit $\lambda = \infty$.

LAGRANGE MULTIPLIERS IN INTRINSIC ELASTICITY

In an intrinsic approach to three-dimensional linearized elasticity, the unknown is the linearized strain tensor field (or equivalently the stress tensor field by means of the constitutive equation), instead of the displacement vector field in the classical approach. We consider here the pure traction problem and the pure displacement problem and we show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations (the form of which depends on the type of boundary conditions considered). Using the Babuška-Brezzi inf-sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints. Such results have potential applications to the numerical analysis and simulation of the intrinsic approach to three-dimensional linearized elasticity.

A Lagrangian approach to intrinsic linearized elasticity

We consider the pure traction problem and the pure displacement problem of three-dimensional linearized elasticity. We show that, in each case, the intrinsic approach leads to a quadratic minimization problem constrained by Donati-like relations. Using the Babuška–Brezzi inf–sup condition, we then show that, in each case, the minimizer of the constrained minimization problem found in an intrinsic approach is the first argument of the saddle-point of an ad hoc Lagrangian, so that the second argument of this saddle-point is the Lagrange multiplier associated with the corresponding constraints.

Existence theorems in intrinsic nonlinear elasticity

We first show how the displacement-traction problem of nonlinear three-dimensional elasticity can be recast either as a boundary value problem or as a minimization problem over a Banach manifold, where the unknown is the Cauchy–Green strain tensor instead of the deformation as is customary. We then consider the pure displacement problem, and we show that, under appropriate smoothness assumptions on the data, either problem recast in this fashion possesses at least a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. In particular, the minimization problem provides an example where the functional is not coercive.

The pure displacement problem in nonlinear three-dimensional elasticity: intrinsic formulation and existence theorems

In this Note, the equations of nonlinear three-dimensional elasticity corresponding to the pure displacement problem are recast either as a boundary value problem, or as a minimization problem, where the unknown is in both cases the Cauchy–Green strain tensor, instead of the deformation as is customary. We then show that either problem possesses a solution if the applied forces are sufficiently small and the stored energy function satisfies specific hypotheses. The second problem provides an example of a minimization problem for a non-coercive functional over a Banach manifold. To cite this article: P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The Convergence of Regularized Minimizers for Cavitation Problems in Nonlinear Elasticity

Consider a nonlinearly elastic body which occupies the region $\Omega \subset {\mathbb{R}} ^m$ ($m=2,3$) in its reference state and which is held in tension under prescribed boundary displacements on $\partial \Omega$. Let ${\bf x}_0 \in \Omega$ be any fixed point in the body. It is known from variational arguments that, for sufficiently large boundary displacements, there may exist discontinuous weak solutions of the equilibrium equations corresponding to a hole forming at ${\bf x}_0$ in the deformed body (this is the phenomenon of cavitation). For each $\epsilon >0$, define the regularized domains $\Omega _\epsilon = \Omega \backslash \overline{B_\epsilon ({\bf x}_0)}$ which contain a preexisting hole of radius $\epsilon >0$ centered on ${\bf x}_0$. Now consider the corresponding mixed displacement/traction problem on $\Omega _\epsilon$ in which the boundary $\partial \Omega$ is subject to the same boundary displacements and the deformed cavity surface (i.e., the image of $\partial B_\epsilon$) is required to be stress-free. It follows from variational arguments that there exists a weak solution ${\bf u} _\epsilon$ of this problem for each $\epsilon >0 $. In this paper we prove convergence of these regularized minimizers ${\bf u} _\epsilon$ in the limit as $\epsilon \rightarrow 0$. In particular, we show that if $\epsilon _n \to 0$, then, passing to a subsequence, ${\bf u} _{\epsilon _n} \rightarrow {\bf u}$, where ${\bf u}$ is a minimizer for the original pure displacement problem on $\Omega$. Finally, we study the effect on cavitation of regularizing the variational problem by introducing a surface energy term which penalizes the formation and growth of cavities.

First-Order System Least Squares for the Stokes and Linear Elasticity Equations: Further Results

First-order system least squares (FOSLS) was developed in [ SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741; SIAM J. Numer. Anal., 35 (1998), pp. 320--335] for Stokes and elasticity equations. Several new results for these methods are obtained here. First, the inverse-norm FOSLS scheme that was introduced but not analyzed in [ SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741] is shown to be continuous and coercive in the L2 norm. This result is shown to hold for pure displacement or pure traction boundary conditions in two or three dimensions, and for mixed boundary conditions in two dimensions. Next, the FOSLS schemes developed in [ SIAM J. Numer. Anal., 35 (1998), pp. 320--335] are applied to the pure displacement problem in planar and spatial linear elasticity byeliminating the pressure variable in the FOSLS formulations of [SIAM J. Numer. Anal., 34 (1997), pp. 1727--1741]. The idea of two-dimensional variable rotation is then extended to three dimensions to make the intervariable coupling subdominant (uniformly so in the Poisson ratio for elasticity). This decoupling ensures optimal (uniform) performance of finite element discretization and multigrid solution methods. It also allows special treatment of the new trace variable, which corresponds to the divergence of velocity in the case of Stokes, so that conservation can be easily imposed, for example. Numerical results for various boundary value problems of planar linear elasticity are studied in a companion paper [SIAM J. Sci. Comput., 21 (2000), pp. 1706--1727].

Two‐level preconditioning of pure displacement non‐conforming FEM systems

This paper is concerned with the pure displacement problem of planar linear elasticity. Our interest is focused on the locking-free FEM approximation of the problem in the case of almost incompressible material. Crouzeix-Raviart linear finite elements are implemented. An optimal order pure algebraic multiplicative two-level preconditioner for the related stiffness matrix is developed. The proposed construction is based on a proper hierarchical basis of the FEM space. It is important to note that the nodal FEM spaces corresponding to successive levels of mesh refinements are not nested for the non-conforming elements under consideration. Local spectral analysis is applied to determine the scaling parameter of the preconditioner as well as to estimate the related constant in the strengthened CBS inequality. The derived estimates are uniform with respect to the Poisson ratio ν ∈ (0, 0.5). A set of numerical tests is presented to illustrate the accuracy of the FEM approximation, and the convergence rate of the two-level PCG method. Copyright © 1999 John Wiley & Sons, Ltd.

Two-level preconditioning of pure displacement non-conforming FEM systems

This paper is concerned with the pure displacement problem of planar linear elasticity. Our interest is focused on the locking-free FEM approximation of the problem in the case of almost incompressible material. Crouzeix-Raviart linear finite elements are implemented. An optimal order pure algebraic multiplicative two-level preconditioner for the related stiffness matrix is developed. The proposed construction is based on a proper hierarchical basis of the FEM space. It is important to note that the nodal FEM spaces corresponding to successive levels of mesh refinements are not nested for the non-conforming elements under consideration. Local spectral analysis is applied to determine the scaling parameter of the preconditioner as well as to estimate the related constant in the strengthened CBS inequality. The derived estimates are uniform with respect to the Poisson ratio ν ∈ (0, 0.5). A set of numerical tests is presented to illustrate the accuracy of the FEM approximation, and the convergence rate of the two-level PCG method. Copyright © 1999 John Wiley & Sons, Ltd.

Block-Triangular Preconditioners for Saddle Point Problems with a Penalty Term

Block-triangular preconditioners for a class of saddle point problems with a penalty term are considered. An important example is the mixed formulation of the pure displacement problem in linear elasticity. It is shown that the spectrum of the preconditioned system is contained in a real, positive interval and that the interval bounds can be made independent of the discretization and penalty parameters. This fact is used to construct bounds of the convergence rate of the GMRES method with respect to an energy norm. Numerical results are given for GMRES and BI-CGSTAB.

A Nonconforming Mixed Multigrid Method for the Pure Displacement Problem in Planar Linear Elasticity

An optimal order multigrid method is developed for the pure displacement problem in two-dimensional linear elasticity. It is based on a nonconforming mixed formulation, where the displacement is approximated by weakly continuous piecewise linear vector functions, and the pressure is approximated by piecewise constants. The full multigrid convergence is proved. The performance of the multigrid method does not deteriorate as the material becomes nearly incompressible.

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation