Conducción de Calor en Placas Matalicas Perforadas. Parte 1: Modelo y Solución Debil en un Paso de Tiempo

Proponemos una formulaci6n dl!bil de un modelo de conducci6n de calor para Ia detecci6n de perforaciones en placas metalicas. Una discretizaci6n impHcita en tiempo produce un sistema leneal acoplado de ecuaciones diferenciales parciales. En cada paso de tiempo, el sistema se reduce a un problema de Helmholtz con condiciones de frontera tipo Robin y demostramos que su formulaci6n debil equivalente es un problema bien planteado.We propose a weak formulation of a heat conduction model for the detection of holes in metallic plates. An implicit discretization in time leads to a cou p l ed , linear system of partial differential equations. At each time step, the system reduces to a Helmholtz problem with Robin boundary conditions and we show that its equivalent weak formulation is a well-posedness problem

Modelo numérico de elementos finitos para la simulación de conducción de calor en placas perforadas de metal y materiales compuestos tipo carbono-carbono

In this work is presented a numerical model based on finite element method (FEM) for the simulation of heat conduction in perforated plates of metal and material composed of carbon fibers. For this, a mathematical model based on a type Robin boundary problem for the Helmholtz operator is proposed for demonstrating that it is a well-posed problem in the distributions sense. The numerical model was obtained through the equivalent variational problem and its discretization using the Courant finite element for a finite element mesh of size h. The numerical approach was of order h with respect to H1 norm and order h2 with respect to norm L2. Numerical simulations through a designed and implemented code in Matlab were done and thus, the profiles of temperatures were visualized for some test problems assuming a regular perforation in metal plates and carbon fiber materials. En este trabajo se presenta un modelo numérico basado en el método de los elementos finitos (FEM) para la simulación de la conducción de calor en placas perforadas de metal y de material compuesto de fibras de carbono. Para esto, se propone un modelo matemático que consistió en un problema de frontera tipo Robín para el operador de Helmholtz y se demuestra que está bien planteado en el sentido de distribuciones. El modelo numérico se obtuvo por medio del problema variacional equivalente y su discretización usando el elemento finito de Courant para un tamaño h de la malla de elementos finitos. La aproximación numérica fue del orden h respecto a la norma H1 y de orden h2 respecto a la norma L2. Se realizaron simulaciones numéricas mediante un código numérico en MATLAB y acá, se visualizaron los perfiles de temperaturas de algunos problemas de prueba en los que se supone una perforación regular en placas de metal y materiales de fibras de carbono

A Banach spaces-based analysis of a new fully-mixed finite element method for the Boussinesq problem

In this paper we propose and analyze, utilizing mainly tools and abstract results from Banach spaces rather than from Hilbert ones, a new fully-mixed finite element method for the stationary Boussinesq problem with temperature-dependent viscosity. More precisely, following an idea that has already been applied to the Navier–Stokes equations and to the fluid part only of our model of interest, we first incorporate the velocity gradient and the associated Bernoulli stress tensor as auxiliary unknowns. Additionally, and differently from earlier works in which either the primal or the classical dual-mixed method is employed for the heat equation, we consider here an analogue of the approach for the fluid, which consists of introducing as further variables the gradient of temperature and a vector version of the Bernoulli tensor. The resulting mixed variational formulation, which involves the aforementioned four unknowns together with the original variables given by the velocity and temperature of the fluid, is then reformulated as a fixed point equation. Next, we utilize the well-known Banach and Brouwer theorems, combined with the application of the Babuška-Brezzi theory to each independent equation, to prove, under suitable small data assumptions, the existence of a unique solution to the continuous scheme, and the existence of solution to the associated Galerkin system for a feasible choice of the corresponding finite element subspaces. Finally, we derive optimal a priori error estimates and provide several numerical results illustrating the performance of the fully-mixed scheme and confirming the theoretical rates of convergence

Analysis of a semi-augmented mixed finite element method for double-diffusive natural convection in porous media

In this paper we study a stationary double-diffusive natural convection problem in porous media given by a Navier-Stokes/Brinkman type system, for describing the velocity and the pressure, coupled to a vector advection-diffusion equation relate to the heat and substance concentration, of a viscous fluid in a porous media with physical boundary conditions. The model problem is rewritten in terms of a first-order system, without the pressure, based on the introduction of the strain tensor and a nonlinear pseudo-stress tensor in the fluid equations. After a variational approach, the resulting weak model is then augmented using appropriate redundant penalization terms for the fluid equations along with a standard primal formulation for the heat and substance concentration. Then, it is rewritten as an equivalent fixed-point problem. Well-posedness results for both the continuous and the discrete schemes are stated, as well as the respective convergence result under certain regularity assumptions combined with the Lax-Milgram theorem, and the Banach and Brouwer fixed-point theorems. In particular, Raviart-Thomas elements of order k are used for approximating the pseudo-stress tensor, piecewise polynomials of degree ≤k and ≤k+1 are utilized for approximating the strain tensor and the velocity, respectively, and the heat and substance concentration are approximated by means of Lagrange finite elements of order ≤k+1. Optimal a priori error estimates are derived and confirmed through some numerical examples that illustrate the performance of the proposed semi-augmented mixed-primal scheme.Agencia Nacional de Investigación y Desarrollo/[Fondecyt 11190241]/ANID/ChileAgencia Nacional de Investigación y Desarrollo/[FB210005]/ANID/ChileUniversidad Nacional de Costa Rica/[0140-20]/UNA/Costa RicaUniversidad de Costa Rica/[540-C0-089]/UCR/Costa RicaUCR::Sedes Regionales::Sede de Occident