PhD Student, Civil and Environmental Engineering Department, Vanderbilt University
we propose a stabilized finite element method for alleviating the artificial compliance issue
Composites are now-a-days widely used in aerospace, naval and mechanical industries to build airplanes and cars. Researchers are concerned about predicting their failure mechanisms, prediction of crack propagation and crack growth issues as their uses are growing day by day. To increase the cost effectiveness and efficiency of this process, we do numerical simulation building a computer model of the actual structure. It has been found that delamination is one of the major failure modes in composites. Delamination of composite materials is commonly modeled using intrinsic cohesive zone models (CZMs), which are generally incorporated into the standard finite element (FE) method through a zero-thickness interface (cohesive) element; however, intrinsic CZMs exhibit numerical instabilities when the cohesive stiffness parameters are assumed to be large relative to the elastic stiffness of the composite material. To address this numerical instability issue, we propose a stabilized finite element method by combining the traditional penalty method with the Nitsche’s method that is equally effective for any specified initial stiffness of the cohesive (traction-separation) law. The key advantage of the proposed method is that it generalizes the Nitsche’s method to any traction-separation law with arbitrary large values of initial stiffness and provides a unified way to treat cohesive fracture problems in a variationally consistent and stable manner. We implemented the stabilized method in the commercial finite element software Abaqus via the user element subroutine and simulated benchmark tests for mode I and mixed-mode delamination in isotropic materials to establish the viability of the approach.
Abstract: Second order buckling theory involves a one-way coupled coupled problem where the stress tensor from a plane stress problem appears in an eigenvalue problem for the fourth order Kirchhoff plate. In this paper we present an a posteriori error estimate for the critical buckling load and mode corresponding to the smallest eigenvalue and associated eigenvector. A particular feature of the analysis is that we take the effect of approximate computation of the stress tensor and also provide an error indicator for the plane stress problem. The Kirchhoff plate is discretized using a continuous/discontinuous finite element method based on standard continuous piecewise polynomial finite element spaces. The same finite element spaces can be used to solve the plane stress problem.
Pub.: 04 Sep '15, Pinned: 28 Jun '17
Abstract: We propose a new finite element method for Helmholtz equation in the situation where an acoustically permeable interface is embedded in the computational domain. A variant of Nitsche’s method, different from the standard one, weakly enforces the impedance conditions for transmission through the interface. As opposed to a standard finite-element discretization of the problem, our method seamlessly handles a complex-valued impedance function ZZ that is allowed to vanish. In the case of a vanishing impedance, the proposed method reduces to the classic Nitsche method to weakly enforce continuity over the interface. We show stability of the method, in terms of a discrete Gårding inequality, for a quite general class of surface impedance functions, provided that possible surface waves are sufficiently resolved by the mesh. Moreover, we prove an a priori error estimate under the assumption that the absolute value of the impedance is bounded away from zero almost everywhere. Numerical experiments illustrate the performance of the method for a number of test cases in 2D and 3D with different interface conditions.
Pub.: 05 Mar '16, Pinned: 28 Jun '17