Finite Element Formulation and Solution of Contact-Impact Problems in Continuum Mechanics - IV

SUMMARYIn the past, the combined finite-discrete element was mostly based on linear tetrahedral finite elements. Locking problems associated with this element can seriously degrade the accuracy of their simulations. In this work an efficient ten-noded quadratic element is developed in a format suitable for the combined finite-discrete element method (FEMDEM). The so-called F-bar approach is used to relax volumetric locking and an explicit finite element analysis is employed. A thorough validation of the numerical method is presented including five static and four dynamic examples with different loading, boundary conditions, and materials. The advantages of the new higher-order tetrahedral element are illustrated when brought together with contact detection and contact interaction capability within a new fully 3D FEMDEM formulation. An application comparing stresses generated within two drop experiments involving different unit specimens called Vcross and VRcross is shown. The Vcross and VRcross units of ∼ 3.5×10 4 kg show very different stress generation implying different survivability upon collision with a deformable floor. The test case shows the FEMDEM method has the capability to tackle the dynamics of complex-shaped geometries and massive multi-body granular systems typical of concrete armour and rock armour layers.

Finite strain, finite rotation quadratic tetrahedral element for the combined finite–discrete element method

Xiang

Munjiza

Latham

2009

A finite element method based on C0‐continuous assumed gradients

Wolff

Bucher

2010

SUMMARYThis article presents an alternative approach to assumed gradient methods in FEM applied to threedimensional elasticity. Starting from nodal integration (NI), a general C0-continuous assumed interpolation of the deformation gradient is formulated. The assumed gradient is incorporated using the principle of Hu-Washizu. By dual Lagrange multiplier spaces, the functional is reduced to the displacements as the only unknowns. An integration scheme is proposed where the integration points coincide with the support points of the interpolation. Requirements for regular finite element meshes are explained. Using this interpretation of NI, instabilities (appearance of spurious modes) can be explained. The article discusses and classifies available strategies to stabilize NI such as penalty methods, SCNI, -FEM. Related approaches, such as the smoothed finite element method, are presented and discussed. New stabilization techniques for NI are presented being entirely based on the choice of the assumed gradient interpolation, i.e. nodal-bubble support, edge-based support and support using tensor-product interpolations. A strategy is presented on how the interpolation functions can be derived for various element types. Interpolation functions for the first-order hexahedral element, the first-order and the second-order tetrahedral elements are given. Numerous examples illustrate the strengths and limitations of the new schemes.

An energy‐stable mixed formulation for isogeometric analysis of incompressible hyperelastodynamics

Marsden

Tao

2019

Summary We develop a mixed formulation for incompressible hyperelastodynamics based on a continuum modeling framework recently developed in the work of Liu and Marsden and smooth generalizations of the Taylor‐Hood element based on nonuniform rational B‐splines (NURBSs). This continuum formulation draws a link between computational fluid dynamics and computational solid dynamics. This link inspires an energy stability estimate for the spatial discretization, which favorably distinguishes the formulation from the conventional mixed formulations for finite elasticity. The inf‐sup condition is utilized to provide a bound for the pressure field. The generalized‐α method is applied for temporal discretization, and a nested block preconditioner is invoked for the solution procedure. The inf‐sup stability for different pairs of NURBS elements is elucidated through numerical assessment. The convergence rate of the proposed formulation with various combinations of mixed elements is examined by the manufactured solution method. The numerical scheme is also examined under compressive and tensile loads for isotropic and anisotropic hyperelastic materials. Finally, a suite of dynamic problems is numerically studied to corroborate the stability and conservation properties.