A 3D mortar method for solid mechanicsPuso, Michael A.
doi: 10.1002/nme.865pmid: N/A
A version of the mortar method is developed for tying arbitrary dissimilar 3D meshes with a focus on issues related to large deformation solid mechanics. Issues regarding momentum conservation, large deformations, computational efficiency and bending are considered. In particular, a mortar method formulation that is invariant to rigid body rotations is introduced. A scheme is presented for the numerical integration of the mortar surface projection integrals applicable to arbitrary 3D curved dissimilar interfaces. Here, integration need only be performed at problem initialization such that coefficients can be stored and used throughout a quasi‐static time stepping process even for large deformation problems. A degree of freedom reduction scheme exploiting the dual space interpolation method such that direct linear solution techniques can be applied without Lagrange multipliers is proposed. This provided a significant reduction in factorization times. Example problems which touch on the aforementioned solid mechanics related issues are presented. Published in 2003 by John Wiley & Sons, Ltd.
From mixed finite elements to finite volumes for elliptic PDEs in two and three dimensionsYounes, Anis; Ackerer, Philippe; Chavent, Guy
doi: 10.1002/nme.874pmid: N/A
The link between Mixed Finite Element (MFE) and Finite Volume (FV) methods applied to elliptic partial differential equations has been investigated by many authors. Recently, a FV formulation of the mixed approach has been developed. This approach was restricted to 2D problems with a scalar for the parameter used to calculate fluxes from the state variable gradient. This new approach is extended to 2D problems with a full parameter tensor and to 3D problems. The objective of this new formulation is to reduce the total number of unknowns while keeping the same accuracy. This is achieved by defining one new variable per element. For the 2D case with full parameter tensor, this new formulation exists for any kind of triangulation. It allows the reduction of the number of unknowns to the number of elements instead of the number of edges. No additional assumptions are required concerning the averaging of the parameter in hetero‐ geneous domains. For 3D problems, we demonstrate that the new formulation cannot exist for a general 3D tetrahedral discretization, unlike in the 2D problem. However, it does exist when the tetrahedrons are regular, or deduced from rectangular parallelepipeds, and allows reduction of the number of unknowns. Numerical experiments and comparisons between both formulations in 2D show the efficiency of the new formulation. Copyright © 2003 John Wiley & Sons, Ltd.
Numerical solution of the Helmholtz equation with high wavenumbersBao, Gang; Wei, G. W.; Zhao, Shan
doi: 10.1002/nme.883pmid: N/A
This paper investigates the pollution effect, and explores the feasibility of a local spectral method, the discrete singular convolution (DSC) algorithm for solving the Helmholtz equation with high wavenumbers. Fourier analysis is employed to study the dispersive error of the DSC algorithm. Our analysis of dispersive errors indicates that the DSC algorithm yields a dispersion vanishing scheme. The dispersion analysis is further confirmed by the numerical results. For one‐ and higher‐dimensional Helmholtz equations, the DSC algorithm is shown to be an essentially pollution‐free scheme. Furthermore, for large‐scale computation, the grid density of the DSC algorithm can be close to the optimal two grid points per wavelength. The present study reveals that the DSC algorithm is accurate and efficient for solving the Helmholtz equation with high wavenumbers. Copyright © 2003 John Wiley & Sons, Ltd.
Implicit plane stress algorithm for multilayer J 2 ‐plasticity using the Prager–Ziegler translation ruleMontáns, Francisco J.
doi: 10.1002/nme.885pmid: N/A
A totally implicit algorithm for plane stress multilayer plasticity is presented. The algorithm presents the plane stress version of the 3D/plane strain model for multilayer plasticity presented in a previous work. As in the 3D/plane strain case, the model is consistent with the principle of maximum dissipation and it may be considered as an extension of classical J2‐plasticity for anisotropic non‐linear kinematic behaviour preserving Masing's rules. In order to obtain the asymptotic second‐order convergence of the Newton algorithm, the consistent tangent moduli are also given. Copyright © 2003 John Wiley & Sons, Ltd.
Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubesArroyo, M.; Belytschko, T.
doi: 10.1002/nme.944pmid: N/A
The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved. Copyright © 2003 John Wiley & Sons, Ltd.