Analysis and computation of a least-squares method for consistent mesh tying

This work presents a new technique for joining dissimilar meshes in non‐linear finite element analysis, with the novel result that the adjoining surfaces do not need to be spatially coincident for patch tests to be passed. The work builds in some sense on results previously presented by Dohrmann et al. (International Journal for Numerical Methods in Engineering 1991; 40:1205), but considerably generalizes upon the results presented there. Most importantly, it is shown that consistency of the tied finite element formulation may be achieved such that implementation of element formulations is not affected, through the use of a three‐field variational principle applied over the contacting region. A key conclusion of the work is that although the approach advocated bears some resemblance to mortar element schemes for contact analysis (see, for example, McDevitt and Laursen, International Journal for Numerical Methods in Engineering 2000; 48:1525), a straightforward application of a mortaring scheme is not sufficient to guarantee patch test passage when dissimilar meshing of curved surfaces leads to local open volumes (or overlaps) between tied surfaces in their reference configurations. The proposed algorithm is demonstrated through a number of numerical examples performed within a matrix‐free non‐linear equation solving framework. Published in 2003 by John Wiley & Sons, Ltd.

Purpose – A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities. Design/methodology/approach – A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation. Findings – Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency. Originality/value – Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.

The Strut and Tie Method (STM) is one effective method for predicting the ultimate strength, failure modes, and behavior of reinforced concrete pile caps. However, the use of the strut and tie method in pile cap design in Indonesia is not as popular as the use of conventional methods. This study aims to evaluate the reinforcement design of pile caps using conventional methods and the finite element method (FEM) approach, utilizing the strut and tie method specified in SNI 2847 2019. The analysis results show that the required area of longitudinal reinforcement in the conventional method does not meet the criteria when evaluated using the STM method. The deficiency in the longitudinal reinforcement area amounts to 40.13% compared to the design results obtained using the STM method. Meanwhile, the use of the FEM method results in a required reinforcement area that exceeds the required area for the STM method. The difference in reinforcement is even greater, reaching up to 7.85% for the FEM method.

Chapter 2 - Galerkin Method of Approximation: Irreducible and Mixed Forms

Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: Compressible and incompressible plasticity

Tire's are made up of a typical fiber–rubber composite structure. A finite element model of a tire with foundation contact is developed in this study. In this model, the rubber compounds are simulated by incompressible elements, which are treated using the Lagrangian multipliers method. The nonlinear mechanical properties of the elastomers are modeled by the Mooney–Rivlin model. The belts, carcass, and bead are modeled by an equivalent orthotropic material model in which the effective moduli are determined from the individual material properties of the rubber compound and cord based on the Halpin–Tsai equations. For composite elements consisting of multi-ply cord–rubber composites, three-dimensional effective elastic constants can be determined using the material model presented by Sun and Li. The contact constraint of the tire with a flat foundation and rigid rim is treated using the variable constraint method. For the large deformation description of the tire, the Lagrangian method is used here. The strain tensor and the stress tensor are selected as, respectively, the Green–Lagrangian strain tensor and the second Piola– Kirchhoff stress tensor. In finite element code, two kinds of three-dimensional elements are used – an eight-node brick isoparametric element (hexahedral element) and a six-node isoparametric element (pentahedral element). The present numerical results show that the reliability and convergence of the model are fairly good. Moreover, three groups of radial tires with different belt widths under inflation and static footprint loading are analyzed using the finite element method. Based on the detailed analysis for stress analysis parameters in the critical regions in the tires, the relative belt edge endurance is predicted.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

A two-grid mixed finite element method for a nonlinear fourth-order reaction–diffusion problem with time-fractional derivative

Coupling Mortar Finite Element and Boundary Element Methods for 2D Navier-Stokes Equations Courant Element: Before and After Iterative Methods for Solving Stiff Elliptic Problems Straight and Curved Finite Elements of Class C1 and Some Applications to Thin Shell Problems Exact Controllability to Solve the Helmholtz Equation with Absorbing Boundary Conditions Cubic Version of FEM in Elliptic Problems with Interfaces and Singularities Least Squares Mixed Finite Elements Turbulence Modelling in Finite Element Industrial Applications Necessary and Sufficient Conditions for the Numerical Approximation of a Partial Differential Equation Depending on a Small Parameter Efficient Solution Methods for Compressible Flow Computations Parallel Finite Volume Algorithms for Solving the Time-Domain Maxwell Equations on Nonstructured Meshes Coupling Between Nonlinear Maxwell and Heat Equations for an Induction Heating Problem: Modelling and Numerical Methods Solving the 3D Harmonic Maxwell Equations with Finite Elements, Lagrange Multipliers, and Iterative Methods Some Applications of the Hierarchic High Order MITC Finite Elements for Reissner-Mindlin Plates Domain Decomposition for Immiscible Displacement in Single Porosity Systems An Error Estimator for Nonconforming Approximations of a Nonlinear Problem Some Observations on Raviart-Thomas Mixed Finite Elements in p Extension for Parabolic Problems Mixed Finite Element Methods in Fluid Structure Systems A Black-Box Solver for the Solution of General Nonlinear Functional Equations by Mixed FEM A Remark on the Asymptotic Behaviour of Parabolic Variational Inequalities and Their Finite Element Approximation by the Courant Element Domain Decomposition vs. Adaptivity Material Optimization of Composites. Part Contents.

This paper presents a new arbitrary Lagrangian–Eulerian (ALE) finite element formulation for finite strain plasticity in non‐linear solid mechanics. We consider the models of finite strain plasticity defined by the multiplicative decomposition of the deformation gradient in an elastic and a plastic part (F = FeFp), with the stresses given by a hyperelastic relation. In contrast with more classical ALE approaches based on plastic models of the hypoelastic type, the ALE formulation presented herein considers the direct interpolation of the motion of the material with respect to the reference mesh together with the motion of the spatial mesh with respect to this same reference mesh. This aspect is shown to be crucial for a simple treatment of the advection of the plastic internal variables and dynamic variables. In fact, this advection is carried out exactly through a particle tracking in the reference mesh, a calculation that can be accomplished very efficiently with the use of the connectivity graph of the fixed reference mesh. A staggered scheme defined by three steps (the smoothing, the advection and the Lagrangian steps) leads to an efficient method for the solution of the resulting equations. We present several representative numerical simulations that illustrate the performance of the newly proposed methods. Both quasi‐static and dynamic conditions are considered in these model examples. Copyright © 2003 John Wiley & Sons, Ltd.

The least-squares finite element method for first order systems corresponding to second order linear two-point boundary value problems is considered. A theoretical analysis and error estimates are developed and the estimates are seen to be consistent with our previous numerical studies in Carey and Shen [5]. The method is not subject to the LBB condition and we consider, in particular, the estimates when the polynomial degree differs for the corresponding variables. Superconvergence estimates are also developed.

Two simple finite element methods for Reissner–Mindlin plates with clamped boundary condition

Having established a basic theoretical and implementational framework from which to view problems in nonlinear mechanics, we begin now with our specific study of contact problems. The problems discussed in the past chapter, while widely representative of issues arising in nonlinear solid mechanics, are also noteworthy in that the boundary conditions required to pose them are in some sense rather idealized. In the framework established thusfar, displacements and tractions that are to be posed on material boundaries are known a priori, as are the areas over which they act. Effectively, when we wish to extend such a framework to encompass potential contact between multiple bodies, we must introduce a new notational framework in which contact areas, contact forces, and motions of associated boundaries are unknown beforehand, and must be determined as part of the solution.

The handling and control of towed cable and body systems onboard surface ships and submarines presents a significant technical challenge to design engineers in the defense and ocean industries. The current approaches rely heavily on the empirical methods and the time-consuming and costly prototype testing. Computer simulation provides a cost effective way to reduce the high risks associated with the towed cable/body system. However, the current dynamic analysis of towed cables is mostly done by the finite difference (FD) method in stead of the finite element (FE) method that is widely used in almost all engineering fields. This paper presents an alternative FE method to simulate the dynamics of towed cable and body system, in which the large rigid body motion is coupled with small elastic deformation. The newly derived FE method is formulated in terms of element nodal positions, which is different from the existing FE methods that use displacements. The alternative FE method solves for the cable position directly in order to eliminate accumulated numerical errors arising from existing FE methods that solve for displacements first in order to obtain the cable position over very long period of time. The alternative FE formulation is implemented and applied to real applications to demonstrate its robustness by comparing simulation results with the experimental and sea trial data.

In general, all kinematic and constitutive relations in vehicle system dynamics are nonlinear. The modeling of such systems therefore leads to nonlinear dynamic systems, which must be investigated with respect to their properties and evolution.