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See detailA coupled electro-thermal Discontinuous Galerkin method
Homsi, Lina ULg; Geuzaine, Christophe ULg; Noels, Ludovic ULg

in Journal of Computational Physics (in press)

This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems. In this paper we discuss the fundamental ... [more ▼]

This paper presents a Discontinuous Galerkin scheme in order to solve the nonlinear elliptic partial differential equations of coupled electro-thermal problems. In this paper we discuss the fundamental equations for the transport of electricity and heat, in terms of macroscopic variables such as temperature and electric potential. A fully coupled nonlinear weak formulation for electro-thermal problems is developed based on continuum mechanics equations expressed in terms of energetically conjugated pair of fluxes and fields gradients. The weak form can thus be formulated as a Discontinuous Galerkin method. The existence and uniqueness of the weak form solution are proved. The numerical properties of the nonlinear elliptic problems i.e., consistency and stability, are demonstrated under specific conditions, i.e. use of high enough stabilization parameter and at least quadratic polynomial approximations. Moreover the prior error estimates in the H1-norm and in the L2-norm are shown to be optimal in the mesh size with the polynomial approximation degree. [less ▲]

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See detailDevelopment of non-linear Electro-Thermo-Mechanical Discontinuous Galerkin formulations
Homsi, Lina ULg

Doctoral thesis (2017)

A coupled Electro-Thermo-Mechanical Discontinuous Galerkin (DG) method is developed considering the non-linear interactions of electrical, thermal, and mechanical fields. The numerical properties of the ... [more ▼]

A coupled Electro-Thermo-Mechanical Discontinuous Galerkin (DG) method is developed considering the non-linear interactions of electrical, thermal, and mechanical fields. The numerical properties of the DG are demonstrated, such as uniqueness, consistency, stability and the optimal convergence rate. The framework is applied to simulate the response of smart composite materials, where the shape memory effect is triggered by the Joule effect. [less ▲]

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See detailUnified treatment of microscopic boundary conditions in computational homogenization method for multiphysics problems
Nguyen, Van Dung ULg; Wu, Ling ULg; Homsi, Lina ULg et al

Conference (2016, September 08)

Computational homogenization (so-called FE2) method is an effective tool to model complex behavior of heterogeneous media allowing direct coupling between the structure response and the evolving ... [more ▼]

Computational homogenization (so-called FE2) method is an effective tool to model complex behavior of heterogeneous media allowing direct coupling between the structure response and the evolving microstructure not only in purely mechanical problems but also in multiphysics problems [1]. The basic idea of this method is to obtain the macroscopic constitutive relationships from the resolution of the microscopic boundary value problem (BVP) defined on a representative volume element. This method does not requires any constitutive assumption at the macroscopic level, but an appropriate microscopic boundary condition has to be defined. Our work focuses on the unified treatment of the microscopic boundary condition in a multiphysics microscopic BVP. In particular, an efficient way to compute the tangent operator is developed for an arbitrary kind of boundary conditions. When considering the FE2method, the homogenized stresses and homogenized tangents at every macroscopic integration points are required. From the energy consistency condition between macroscopic and microscopic problems, the homogenized stresses can be easily computed by the volumetric averaging integrals of the microscopic counterparts. The required homogenized tangents often follows a stiffness condensation from the microscopic stiffness matrix at the equilibrium state [2]. When using the stiffness condensation, the microscopic stiffness matrix needs to be partitioned, and dense matrices based on Schur complements (under a matrix form 𝐊̃ 𝑏𝑏=𝐊𝑏𝑏−𝐊𝑏𝑖𝐊𝑖𝑖−1𝐊𝑖𝑏) have to be estimated. The matrix operations based on Schur complements require a large time consuming and a lot of memory when increasing the number of degrees of freedom of the microscopic BVPs. This work proposes an efficient method allowing to compute the homogenized tangents without significant effort. The microscopic stiffness matrix does not need to be partitioned. The homogenized tangents are computed by solving a linear system, which is based on the linearized system at the converge solution of the microscopic BVP, with multiple right hand sides. With proposed numerical improvements, the FE2 method is used in a fully thermo-mechanically-coupled simulation. The temperature-dependent elastoplastic behavior, thermal conduction as well as the heat conversion from the mechanical deformation are considered in the hyperelastic large strain framework. [1]. Geers, M. G. D., Kouznetsova, V. G., Brekelmans, W. A. M., 2010. J. Comput. Appl. Math. 234 (7), 2175-2182. [2]. Kouznetsova, V., Brekelmans, W. A. M., Baaijens, F. P. T., 2001. Comput. Mech. 27 (1), 37-48. [less ▲]

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See detailA coupled electro-thermo-mechanical discontinuous Galerkin method applied on composite materials
Homsi, Lina ULg; Noels, Ludovic ULg

Conference (2016, September 07)

Carbon fiber reinforced polymer composites have become increasingly important due to their unique properties which are appreciated in many practical applications such as low weight, low cost, low density ... [more ▼]

Carbon fiber reinforced polymer composites have become increasingly important due to their unique properties which are appreciated in many practical applications such as low weight, low cost, low density, high mechanical characteristics. Moreover the range of their electrical conductivity can be controlled by the amount of carbon fibers. Carbon fiber reinforced polymer composites consist of at least two components, a polymer matrix (generally dielectric) and electrically conductive fillers. This combination results in multifunctional composites, both structural and conductive. The existence of the polymer matrix will avoid catastrophic failure due to fiber breaking, and the existence of the carbon fibers will enhance strength and stiffness on one hand, and will allow to a significant temperature gradient when electric current is applied on the other hand. The objective of this paper is to study the response of the carbon fiber reinforced polymer composites when an electric power is applied and to determine the effective properties. To this end governing equations describing electro-thermo-mechanical coupling in composite materials are developed and discretized using the Discontinuous Galerkin (DG) finite element method. DG methods have many advantages such as optimal convergence and local approximation properties in addition to their flexibility for mesh adaption and their straightforward use of high order polynomial approximations. A micromechanical model of unidirectional carbon fibers dispersed in a polymer matrix is formulated considering the interaction of electrical, thermal and mechanical fields It is then solved using the DG method to determine the time dependent response of the electro-thermo-mechanical coupling and quantify the variation of the fields. [less ▲]

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