A continuous sensitivity equation method for conduction and phase change problems

Abstract

This paper presents a Continuous Sensitivity Equation (CSE) that describes the sensitivity of temperature with respect to shape parameters, physical properties and other parameters. We begin by introducing the notion of sensitivities using a one-dimensional, steady-state conduction problem for a composite rod. The model is extended to incorporate contact resistance at the interface between different materials. These examples illustrate some of the difficulties involved in computing sensitivities: accurate evaluation of boundary conditions (for shape sensitivities) and the presence of discontinuities in sensitivities across material interfaces. Solution methods introduced for the above problems are e&.end.4 cr. c---c CLC’L-GbU Vv Ill~a~ ~IIC 0ce~an problem, where we derive a finite element method based on an enthalpy formulation to solve the combined temperature and sensitivity equations. Introduction Sensitivity variables are used in a wide range of engineering prob1ems.l Applications include optimal design, parameter estimation, uncertainty analysis,2 computing rate derivatives,3 and sensitivity *Professor, Associate Fellow AIAA tAssistant Professor, Senior Member AIAA %search Officer Copyright @ 2000 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. studies of engineering systems112v4 For the present study, we define sensitivities as the partial derivatives of field variables (temperature) with respect to model parameters (physical properties, boundary conditions, shape parameters or any other coefficients appearing in the model). Dowding et a1.5 and Blackwell et a1.6 have presented sensitivity equations for conduction problems. Their work is restricted to value sensitivities and does not cover sensitivities to shape parameters. Their development is performed on the integral form used in the finite volume method. While, efficient and elegant for value sensitivities, their approach leads to complications when trying to apply this for shape sensitivities. Although there are many approaches for computing sensitivity variables7-g we emphasiy,e j&s con&-do-us sensitivity equation (@SE) approach. 7y10-12 The sensitivity equations are derived directly from the differential equations thus making the approach simple for both value and shape sensitivities. This paper presents the development of the Continuous Sensitivity Equation (CSE) Method for conduction and phase change problems, The development is performed for value and shape sensitivities, and follows the approach presented by Borggaard et a1.7r lo The development highlights characteristics of the CSE and provides the grounds to identify numerical problems that arise due to these specific characteristics. The resulting SEM equations are 1 American Institute of Aeronautics and Astronautics (c)2000 American Institute of Aeronautics & Astronautics or published with permission of author(s) and/or author(s)’ sponsoring organization. Fig. 1 A Two Material Composite Rod solved by a finite element method specifically designed to handle their numerical approximation. The paper is organized as follows. We first discuss two simple problems of heat conduction in a composite wall. The first one represents a perfectly conducting interface. The second variant includes contact effects between the two materials. We consider sensitivities with respect to boundary temperatures, material conductivities and the interface location. The first two cases are examples of so-called value sensitivities while the last one is a typical example of shape sensitivities. The closed-form solution provides insight into the nature of sensitivities and the potential difficulties encountered in their numerical solution. The differential equations and boundary conditions for the sensitivities are then formally de rived. The analysis is extended to the Stefan problem modeling solidification. A closed-form solution to the iso-thermal phase change problem identifies serious potential numerical difficulties. The sensitivity equation is then derived and an enthalpy based formulation is presented that is applicable to both the Stefan problem and its sensitivity equations. Analytical Sensitivities for Conduction Consider the problem of finding the temperature profile in a composite rod as shown in Figure 1. The rod lies between xi and x2, while the temperatures at the respective ends are maintained at Tl and Tz. The rod is constructed of two materials with their interface located at zi E (zi, ~2). The material conductivity of the rod is ki in (51, xi) and JQ in (zi, ~2). Presently, we assume that the materials are in perfect contact. In later discussions, we assume there is a contact resistance. Composite Rod with Perfect Contact The temperature in the rod is a function of z that depends on all seven design parameters, i.e. where we separate the independent variables from the problem parameters with a semicolon. The temperature sensitivities are partial derivatives of the temperature with respect to the parameters (in this case, Tl, Tz, kl, kz, xi, xi and 52). The notation