Locking-Proof Tetrahedra

The simulation of incompressible materials suffers from locking when using the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio gets close to 0.5 and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating incompressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure, and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young’s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean, and we focus on adapting it to our method. For , we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking, which, in turn, affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis.

Contact interaction of a predeformed plate which lies without friction on rigid base with a parabolic indenter

Within the framework of linearized formulation of a problem of the elasticity theory, the stress-strain state of a predeformed plate, which is modeled by a prestressed layer, is analyzed in the case of its smooth contact interaction with a rigid axisymmetric parabolic indenter. The dual integral equations of the problem are solved by representing the quested-for functions in the form of a partial series sum by the Bessel functions with unknown coefficients. Finite systems of linear algebraic equations are obtained for determination of these coefficients. The influence of the initial strains on the magnitude and features of the contact stresses and vertical displacements on the surface of the plate is analyzed for the case of compressible and incompressible solids. In order to illustrate the results, the cases of the Bartenev – Khazanovich and the harmonic-type potentials are addressed.

A Concept of Cell-Based Smoothed Finite Element Method Using 10-Node Tetrahedral Elements (CS-FEM-T10) for Large Deformation Problems of Nearly Incompressible Solids

A new concept of smoothed finite element method (S-FEM) using 10-node tetrahedral (T10) elements, CS-FEM-T10, is proposed. CS-FEM-T10 is a kind of cell-based S-FEM (CS-FEM) and thus it smooths the strain only within each T10 element. Unlike the other types of S-FEMs [node-based S-FEM (NS-FEM), edge-based S-FEM (ES-FEM), and face-based S-FEM (FS-FEM)], CS-FEM can be implemented in general finite element (FE) codes as a piece of the element library. Therefore, CS-FEM-T10 is also compatible with general FE codes as a T10 element. A concrete example of CS-FEM-T10 named SelectiveCS-FEM-T10 is introduced for large deformation problems of nearly incompressible solids. SelectiveCS-FEM-T10 subdivides each T10 element into 12 four-node tetrahedral (T4) subelements with an additional dummy node at the element center. Two types of strain smoothing are conducted for the deviatoric and hydrostatic stress evaluations and the selective reduced integration (SRI) technique is utilized for the stress integration. As a result, SelectiveCS-FEM-T10 avoids the shear/volumetric locking, pressure checkerboarding, and reaction force oscillation in nearly incompressible solids. In addition, SelectiveCS-FEM-T10 has a relatively long-lasting property in large deformation problems. A few examples of large deformation analyses of a hyperelastic material confirm the good accuracy and robustness of SelectiveCS-FEM-T10. Moreover, an implementation of SelectiveCS-FEM-T10 in the FE code ABAQUS as a user-defined element (UEL) is conducted and its capability is discussed.

A Locking-Free Face-Based S-FEM via Averaging Nodal Pressure using 4-Nodes Tetrahedrons for 3D Explicit Dynamics and Quasi-statics

A locking-free face-based S-FEM, combined with the Averaging Nodal Pressure (ANP) technique, is proposed to solve explicit dynamics of geometric nonlinear nearly-incompressible solids, using simplest linear tetrahedral elements (FS-FEM/ANP-T4). An explicit Adaptive Dynamic Relaxation (ADR) technique is also implemented for the analysis of quasi-static problems. Our studies have found that the proposed method has better accuracy and convergence compared to the standard FEM with ANP (FEM/ANP) and previous selective face-based and Node-based S-FEM (FS/NS-FEM). With the ADR, proposed method can reach the nonlinear quasi-static response much faster than the conventional explicit dynamic relaxation. No temporal instability is observed in FS-FEM/ANP-T4 in large deformation case. In addition, FS-FEM/ANP-T4 also equips the robustness against mesh distortion as FS/NS-FEM but uses less computational time. It has also been applied to solve a practical 3D problem, a rubber hanger for the car exhaust system. FS-FEM/ANP-T4 can be considered as an excellent numerical method other than FS/NS-FEM for simulating rubber-like materials.