More on Equivalent Formulation of Implicit Complementarity Problem

This article presents an equivalent formulation of the implicit complementarity problem. We demonstrate that solution of the equivalent formulation is equivalent to the solution of the implicit complementarity problem. Moreover, we provide another equivalent formulation of the implicit complementarity problem using a strictly increasing function

A two-step iteration method for solving vertical nonlinear complementarity problems

In this paper, for vertical nonlinear complementarity problems, a two-step modulus-based matrix splitting iteration method is established by applying the two-step splitting technique to the modulus-based matrix splitting iteration method. The convergence theorems of the proposed method are given when the number of system matrices is larger than 2. Numerical results show that the convergence rate of the proposed method can be accelerated compared to the existing modulus-based matrix splitting iteration method

The error and perturbation bounds for the absolute value equations with some applications

To our knowledge, so far, the error and perturbation bounds for the general absolute value equations are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error bounds and perturbation bounds for two types of absolute value equations (AVEs): Ax-B|x|=b and Ax-|Bx|=b. Some useful error bounds and perturbation bounds for the above two types of absolute value equations are presented. By applying the absolute value equations, we also obtain the error and perturbation bounds for the horizontal linear complementarity problem (HLCP). In addition, a new perturbation bound for the LCP without constraint conditions is given as well, which are weaker than the presented work in [SIAM J. Optim., 2007, 18: 1250-1265] in a way. Besides, without limiting the matrix type, some computable estimates for the above upper bounds are given, which are sharper than some existing results under certain conditions. Some numerical examples for the AVEs from the LCP are given to show the feasibility of the perturbation bounds

Locking-Proof Tetrahedra

The simulation of incompressible materials suffers from locking when using the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio gets close to 0.5 and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating incompressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure, and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young\u27s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean, and we focus on adapting it to our method. For , we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking, which, in turn, affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis