Smart computational fracture of materials and structures

Advances in Adaptive Computational Methods in Mechanics

One of the major research issues in adaptive finite element analysis is the feedback control system used to guide the adaption. Essentially, one needs to resolve which error data to feedback after each iteration, and how to use it to initialize the next adaptive step. Variational aspects of optimal discretizations for scalar Poisson and Helmholtz systems are used to derive new refinement criteria for adaptive finite element solvers. They are shown to be effective and economical for h-, p- and hp-schemes.

Two mesh refinement indicators based on the gradients of effective stress (GSIG) and effective plastic strain (GEPS), respectively, are proposed for adaptive finite element analysis of the large deformation, quasi‐static or dynamic response of shell structures. The mesh refinement indicators are based on equi‐distributing the variation of stresses or plastic strains over the elements of the mesh. A program module is developed and implemented in the non‐linear explicit finite element code LS‐DYNA. This module provides element‐wise refinement evaluations so that selective mesh refinements are carried out in regions of the mesh where the values of local indicators exceed a user‐specified tolerance. The FE model of a conventional deep drawing process is used as a numerical model, including both material and geometrical non‐linearities, in order to demonstrate the versatility of the two refinement indicators. Four different refinement indicators, based on angle change, thickness change, GSIG and GEPS, are applied in this investigation. The numerical results are compared with experimental results regarding the thickness distribution versus cup height, cup height variation versus circumference angle, effective plastic strain in the deformed sheet and punch force. It is shown that the proposed indicators can identify finite elements which have high gradients of effective stress or effective plastic strain so that the mesh is refined in the regions undergoing the most severe deformations and the numerical results are improved.

Adaptive mesh refinement has the potential of making the finite element computation of magnetic field problems completely automatic. In adaptive procedures, the field problem is solved iteratively, beginning with a coarse mesh and refining it in locations of greatest error. Methods of mesh refinement for triangular finite element grids are surveyed and the use of local error estimates in the adaptive process is described. It is concluded that the Delaunay triangulation provides the best method of mesh refinement, while complementary variational principles provide accurate error bounds on the solution.

3D adaptive finite element modeling of non-planar curved crack growth using the weighted superconvergent patch recovery method

A plasticity model with yield limit degradation is implemented in a boundary element program to study the fracture behavior of quasi-brittle materials. A special integration method is applied to deal with the singular integrations encountered in the volume integrals over internal cells. Strain-softening damage localizations are investigated. It is found that the damage tends to localize into a zone of one cell wide which leads to an incorrect result when different cell meshes are used in the analysis. Some sort of localization limiter has to be incorporated into the analysis in order to achieve meaningful results. INTRODUCTION In the simulation of fracture behavior of quasi-brittle materials such as concrete or ceramics, it is essential to take into account the evolution of a relative large fracture process zone forms in front of the crack tip. The process zone consists of hundreds of microcracks and its overall behavior can generally be simulated either by discrete-cracking models such as the fictitious crack model [4, 5] or smeared-cracking models such as strain-softening damage models [2, 3]. In this paper the fracture process zone is simulated by a plasticity model with yield limit degradation. When the stresses near the crack tip reach the yield surface, the yield surface will contract rather than expand. This results in a decrease in stresses during an increase in strains, i.e., a strain softening. The decrease in stresses simulates a material damage caused by mainly the formation of microcracks. Difficulty has been encountered in the finite element analysis involving strain-softening behavior. As different meshes are used in an analysis, Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X 304 Boundary Elements different results are obtained. The results are not objective with respect to finite element meshes. Also, the energy dissipation of a structure at failure decreases with the refinement of the finite element mesh. It implies that the structure will fail at near zero energy dissipation when a very fine mesh is used. This is in contradiction with the reality. All these problems are considered due to damage localization [1,2,3]. In this paper, the plasticity model with yield limit degradation is incorporated into a boundary element program to study the damage localization problems. Many publications on the boundary element analysis of elastoplastic problems [7-14] have been seen since the first paper on the problem published in 1971 by Swedlow and Cruse [6]. However, the literature on the study of damage localization problems using boundary element approach is rarely seen. The initial stress boundary element method is applied to deal with the nonlinear damage problems, For localization problems, damage tends to localize into a small region of the body, and thus the internal cells required for integration purposes are needed only in a small region rather than the entire body. A special semi-analytical integration method for triangular cell is used to carry out the volume integration over the cell that contains a singular node. A rectangular panel with three different cell meshes are analyzed. It is shown that the results are not objective to the cell meshes even though all other conditions are kept the same. BOUNDARY ELEMENT FORMULATION FOR ELASTOPLASTIC ANALYSIS For elastoplastic problems based on small strain theory, the basic equations in incremental form are as follows: The governing equations are da^ + db* = 0 (1) with boundary conditions du;(x) = (x) x e I (2) cR (x) x e I I + T^T and total strain increment de^ = de^ 4de^= (du + dUjJ/2 (3) where de^ and de^ are elastic and plastic strain increments. T is the boundary of body 0 under consideration, da^ ,db; ,du;, dr; are the stress, body force (per unit volume), displacement, and traction increment respectively. du ± and df j are prescribed displacements and tractions on boundary I and r%. Transactions on Modelling and Simulation vol 2, © 1993 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Elements 305 The yield criterion is F(a%,k)= f(a$ <Kk) = 0 (4) where F and f are yield functions, $ is a work hardening function, and k is the work hardening parameter. The constitutive equations are de'u) = d<^ do, = DV^U (5) where D^= &pvl(l-2v)] 6fo + M<Wji + W

Results of the application of an adaptive finite element (FE) based solution using the FETK library of M. Holst to Density Functional Theory (DFT) approximation to the electronic structure of atoms and molecules are reported. The severe problem associated with the rapid variation of the electronic wave functions in the near singular regions of the atomic centers is treated by implementing completely unstructured simplex meshes that resolve these features around atomic nuclei. This concentrates the computational work in the regions in which the shortest length scales are necessary and provides for low resolution in regions for which there is no electron density. The accuracy of the solutions significantly improved when adaptive mesh refinement was applied, and it was found that the essential difficulties of the Kohn-Sham eigenvalues equation were the result of the singular behavior of the atomic potentials. Even though the matrix representations of the discrete Hamiltonian operator in the adaptive finite element basis are always sparse with a linear complexity in the number of discretization points, the overall memory and computational requirements for the solver implemented were found to be quite high. The number of mesh vertices per atom as a function of the atomic number Z andmore » the required accuracy e (in atomic units) was esitmated to be v (e;Z) = 122:37 * Z2:2346 /1:1173 , and the number of floating point operations per minimization step for a system of NA atoms was found to be 0(N3A*v(e,Z0) (e.g. Z=26, e=0.0015 au, and NA=100, the memory requirement and computational cost would be ~0.2 terabytes and ~25 petaflops). It was found that the high cost of the method could be reduced somewhat by using a geometric based refinement strategy to fix the error near the singularities.« less

This paper presents a novel numerical method for simulating the transport and dispersion of pollutants in the Mediterranean sea. The governing mathematical equations consist of a barotropic ocean model with friction terms, bathymetric forces, Coriolis and wind stresses coupled to an advection–diffusion equation with anisotropic dispersion tensor and source terms. The proposed numerical solver uses a multilevel adaptive semi-Lagrangian finite element method that combines various techniques, including the modified method of characteristics, finite element discretization, coupled projection scheme based on a rotational pressure correction algorithm, and an adaptive L2-projection. The approach employs the gradient of the concentration as an error indicator for enrichment adaptations and increasing the number of quadrature points where needed without refining the mesh. The method is shown to provide accurate and efficient simulations for pollution transport in the Mediterranean sea. The proposed approach distinguishes itself from the well-established adaptive finite element methods for incompressible viscous flows by retaining the same structure and dimension of linear systems during the adaptation process.

A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes

The finite-element analysis is widely used in design stage of electromagnetic apparatuses. The analysis accuracy depends on the characteristics of the finite-element mesh, e.g., number of nodes, number of elements and shape of elements. Recently, the adaptive finite-element analysis is one of the most promising numerical analysis techniques. In process of the adaptive finite-element method, the error evaluation is one of the important schemes. In this paper, a new error evaluation scheme, which is suitable for electromagnetic problems, is proposed. The proposed error evaluation method is then applied to two-dimensional and three-dimensional magnetostatic field problems for its verification

eXtended variational quasicontinuum methodology for lattice networks with damage and crack propagation

This paper analyzes the rate of convergence of the h-p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain \(\Omega_1\) and with constant coefficients in the exterior domain \(\Omega_2 = {\Bbb R}^3 \backslash \overline{\Omega}_1\). This procedure uses the variational formulation of the differential equation in \(\Omega_1\) and involves integral operators on the interface between \(\Omega_1\) and \(\Omega_2\). The finite elements are used to obtain approximate solutions of the differential equation in \(\Omega_1\) and the boundary elements are used to obtain approximate solutions of the integral equations. For given piecewise analytic data we show that the Galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h-p version is used both for finite elements and boundary elements.

Adaptive finite element analysis of stiffened shells

A finite element–boundary element method for advection–diffusion problems with variable advective fields and infinite domains

An original a posteriori error estimator for the Boundary Element Method (BEM) in 3D electromagnetics is presented. Based on this error estimation, an automatic mesh adaptation procedure with regularization is derived. Applications to typical EMC problems show the convergence of the estimator as well as 3D EM fields. Numerical experiments show also that the adaptive procedure is robust and could start with a very coarse mesh.