Stabilized and Modified Mixed u/p Finite Element Formulation for Incompressible Elastic Media using Equal-Order Triangular Elements - Ventura, Giancarlo P.; Zarco, Mark Albert H. (Philippine Engineering Journal; Vol 45 No 02 (2024), 2024)
Abstract – The finite element method is a powerful computational technique used to solve complex problems. However, the method encounters limitations when dealing with incompressible media represented by a Poisson's ratio equal to 0.5. Because of this, mesh locking occurs as certain elements of matrices blow up as others disappear due to the nature of incompressible elasticity that leads to stress oscillations and inaccurate deformation behaviors. This study addresses these challenges by developing a stabilized and modified mixed formulation utilizing equal-order triangular elements. The proposed modified mixed formulation incorporates both displacements and pressures as degrees of freedom by treating them independently, and decoupling stresses and strains by decomposing them into their volumetric and deviatoric parts. Furthermore, the Beltrami-Michell equations are incorporated as a set of compatibility equations to close the boundary value problem. A stabilizing term based on Galerkin mixed methods is introduced to maintain the well-posedness of the problem, addressing issues such as ill-conditioning of the stiffness matrix and preventing spurious stresses. The study also emphasizes the efficiency of unstructured lower order triangular meshes, which, when stabilized, offer a balance between computational cost and accuracy. Since there is limited to no literature regarding the application of such methods in an axisymmetric case, a representative problem in the form of an infinite half-space loaded by a circular footing is presented and validated to demonstrate the effectiveness of the proposed method. The example highlights the formulation's ability to provide accurate results, validating its robustness and practical applicability in various engineering scenarios.   Keywords : finite element method, incompressible elasticity, mixed formulation, triangular elements