A Hybridized Weak Galerkin Finite Element Scheme for the Stokes Equations

In this paper a hybridized weak Galerkin (HWG) finite element method for solving the Stokes equations in the primary velocity-pressure formulation is introduced. The WG method uses weak functions and their weak derivatives which are defined as distributions. Weak functions and weak derivatives can be approximated by piecewise polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. A Lagrange multiplier is introduced to provide a numerical approximation for certain derivatives of the exact solution. With this new feature, HWG method can be used to deal with jumps of the functions and their flux easily. Optimal order error estimate are established for the corresponding HWG finite element approximations for both {\color{black}primal variables} and the Lagrange multiplier. A Schur complement formulation of the HWG method is derived for implementation purpose. The validity of the theoretical results is demonstrated in numerical tests.Comment: 19 pages, 4 tables,it has been accepted for publication in SCIENCE CHINA Mathematics. arXiv admin note: substantial text overlap with arXiv:1402.1157, arXiv:1302.2707 by other author

The method of mothers for non-overlapping non-matching DDM

In this paper we introduce a variant of the three-field formulation where we use only two sets of variables. Considering, to fix the ideas, the homogeneous Dirichlet problem for the Laplace operator in a bounded domain, our variables are: 1) an approximation of the solution on the skeleton (the union of the interfaces of the sub-domains) on an independent grid (that could often be uniform), and 2) the approximations of the solution in each sub-domain, each on its own grid. The novelty is in the way to derive, from the approximation on the skeleton, the values of each trace of the approximations in the subdomains. We do it by solving an auxiliary problem, that resembles the mortar method but is more flexible. Under suitable assumptions, quasi-optimal error estimates are proved, uniformly with respect to the number and size of the subdomains

A variational framework for mathematically nonsmooth problems in solid and structure mechanics

This dissertation presents a new paradigm for addressing multi-physics problems with interfaces in the field of Additive Manufacturing and the modeling of fibrous composite materials. The unique process of adding the material layer by layer in the AM techniques raises the issue about the stability of the interfaces between the layers and along the boundaries of multi-constituent materials. A stabilized interface formulation is developed to model debonding in monotonic loading, fatigue effects in cyclic loading, and thermal effects at interfaces which severely impact the functional life of those materials and structures. The formulation is based on embedding Discontinuous Galkerin (DG) ideas in a Continuous Galerkin (CG) framework. Starting from a mixed method incorporating the Lagrange multiplier along the interface, a pure displacement formulation is derived using the Variational Multiscale Method (VMS). From a mathematical and computational perspective, the key factor influencing the accuracy and robustness of the interface formulation is the design of the numerical flux and the penalty or stability terms. Analytical expressions that are free from user-defined parameters are naturally derived for the numerical flux and stability tensor which are functions of the evolving geometric and material nonlinearity. The proposed framework is extended for debonding at finite strains across general bimaterial interfaces. An interfacial gap function is introduced that evolves subject to constraints imposed by opening and/or sliding interfaces. An internal variable formalism is derived together with the notion of irreversibility of damage results in a set of evolution equations for the gap function that seamlessly tracks interface debonding by treating damage and friction in a unified way. Tension debonding, compression damage, and frictional sliding are accommodated, and return mapping algorithms in the presence of evolving strong discontinuities are developed. This derivation variationally embeds the interfacial kinematic models that are crucial to capturing the physical and mathematical properties involving large strains and damage. The framework is extended for monolithic coupling of thermomechanical fields in the class of problems that have embedded weak and strong discontinuities in the mechanical and thermal fields. Since the derived expressions are a function of the mechanical and thermal fields, the resulting stabilized formulation contains numerical flux and stability tensors that provide an avenue to variationally embed interfacial kinetic and kinematic models for more robust representation of interfacial physics. Representative numerical tests involving large strains and rotations, damage phenomena, and thermal effects are performed to confirm the robustness and accuracy of the method. Comparison of the results with both experimental and numerical results from literature are presented.Ope

Stabilization of Galerkin methods and applications to domain decomposition

Consiglio Nazionale delle Ricerche (CNR). Biblioteca Centrale / CNR - Consiglio Nazionale delle RichercheSIGLEITItal