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Register a new userA Locking-free DP-Q2-P1 MFEM for Incompressible Nonlinear Elasticity Problems
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A mixed finite element method (MFEM), using dual-parametric piecewise bi-quadratic and affine (DP-Q2-P1) finite element approximations for the deformation and the pressure like Lagrange multiplier respectively, is developed and analyzed for the numerical computation of incompressible nonlinear elasticity problems with large deformation gradient, and a damped Newton method is applied to solve the resulted discrete problem. The method is proved to be locking free and stable. The accuracy and efficiency of the method are illustrated by numerical experiments on some typical cavitation problems.
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A mixed finite element method combining an iso-parametric $Q_2$-$P_1$ element and an iso-parametric $P_2^+$-$P_1$ element is developed for the computation of multiple cavities in incompressible nonlinear elasticity. The method is analytically proved to be locking-free and convergent, and it is also shown to be numerically accurate and efficient by numerical experiments. Furthermore, the newly developed accurate method enables us to find an interesting new bifurcation phenomenon in multi-cavity growth.
Intrinsic nonlinear elasticity deals with the deformations of elastic bodies as isometric immersions of Riemannian manifolds into the Euclidean spaces (see Ciarlet [9,10]). In this paper, we study the rigidity and continuity properties of elastic bodies for the intrinsic approach to nonlinear elasticity. We first establish a geometric rigidity estimate for mappings from a Riemannian manifold to a sphere (in the spirit of Friesecke--James--M{u}ller [20]), which is the first result of this type for the non-Euclidean case as far as we know. Then we prove the asymptotic rigidity of elastic membranes under suitable geometric conditions. Finally, we provide a simplified geometric proof of the continuous dependence of deformations of elastic bodies on the Cauchy--Green tensors and second fundamental forms, which extends the Ciarlet--Mardare theorem in [17] to arbitrary dimensions and co-dimensions.
We show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, in a general domain we can treat the Navier-Stokes equations as a perturbed vector diffusion equation, instead of as a perturbed Stokes system. To illustrate the advantages of this view, we provide a simple proof of the unconditional stability of a difference scheme that is implicit only in viscosity and explicit in both pressure and convection terms, requiring no solution of stationary Stokes systems or inf-sup conditions.
We propose a family of mixed finite element that is robust for the nearly incompressible strain gradient model, which is a fourth order singular perturbation elliptic system. The element is similar to the Taylor-Hood element in the Stokes flow. Using a uniform stable Fortin operator for the mixed finite element pairs, we are able to prove the optimal rate of convergence that is robust in the incompressible limit. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical prediction.
We study the inverse electrostatic and elasticity problems associated with Poisson and Navier equations. The uniqueness of solutions of these problems is proved for piecewise constant electric charge and internal stress distributions having a checkered structure: they are constant on rectangular blocks. Such distributions appear naturally in practical applications. We also discuss computational challenges arising in the numerical implementation of our method.
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