International Journal for Numerical Methods in Engineering

Yuan, Rui; Singh, Sudhanshu S.; Chawla, Nikhilesh; Oswald, Jay

doi: 10.1002/nme.4619pmid: N/A

We present two methods for initializing distance functions on adaptively‐refined finite element meshes to represent complex material microstructures from segmented x‐ray tomographic data. Implicit microstructure representation combined with the extended FEM allows modelers to represent complex material microstructures with consistent mesh quality and accuracy. In the first method, a level set evolution equation is formulated and solved by the Galerkin method on an adaptively‐refined mesh. We show that the convergence and stability of this method is optimal for the order of elements used. In the second approach, we initialize the distance field by the fast marching method on a uniform grid, and then project the solution onto the finite element mesh by least‐squares. We show that this latter approach is superior in speed and accuracy. As an example problem, both methods are demonstrated in the initialization of distance fields for two inclusion phases within a Al‐7075 alloy. Copyright © 2014 John Wiley & Sons, Ltd.

Oberrecht, S.P.; Novák, J.; Krysl, P.

doi: 10.1002/nme.4621pmid: N/A

Anisotropic elastic materials, such as the homogenized model of a fiber‐reinforced matrix, can display near rigidity under certain applied stress–the resulting strains are small compared with the strains that would occur for other stresses of comparable magnitude. The anisotropic material could be rigid under hydrostatic pressure if the material were incompressible, as in isotropic elasticity, but also for other stresses.

Mostafa, M.; Sivaselvan, M.V.

doi: 10.1002/nme.4627pmid: N/A

We discuss a strategy to construct corotated frames for three‐dimensional continuum finite elements by minimizing deformations within the frame. We find that irrespective of the type of element and the number of nodes, using a quaternion parametrization of rotations, this minimization is naturally stated as computing the smallest eigenvalue of a 4 × 4 matrix. The simplicity of this smallest eigenvalue plays a crucial role when linearizing the kinematics. Ensuant quaternion algebra, although lengthy, results in remarkably simple formulas for projections that arise in the linearized kinematics. The exact stiffness matrix does not require computation of the second derivative of the rotation function and is also given by a simple formula. As a result, the implementation of this corotational formulation becomes particularly straightforward. Furthermore, in contrast to other results in the literature, the stiffness matrix for elements with translational DOFs is symmetric. For illustration, this corotational formulation is applied to a solid‐shell finite element, and numerical results are presented. Copyright © 2014 John Wiley & Sons, Ltd.

Banks, H. T.; Birch, Malcolm J; Brewin, Mark P; Greenwald, Stephen E; Hu, Shuhua; Kenz, Zackary R; Kruse, Carola; Maischak, Matthias; Shaw, Simon; Whiteman, John R

doi: 10.1002/nme.4631pmid: 25834284

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685–6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space‐time coupling matrices are diagonalizable over C for r ⩽100, and this means that the time‐coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG‐in‐time methodology, for the first time, to second‐order wave equations including elastodynamics with and without Kelvin–Voigt and Maxwell–Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high‐order (up to degree 7) temporal and spatio‐temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.