Optimal design of solidification processes

An optimal design algorithm is presented for the analysis of general solidification processes, and is demonstrated for the growth of GaAs crystals in a Bridgman furnace. The system is optimal in the sense that the prespecified temperature distribution in the solidifying materials is obtained to maximize product quality. The optimization uses traditional numerical programming techniques which require the evaluation of cost and constraint functions and their sensitivities. The finite element method is incorporated to analyze the crystal solidification problem, evaluate the cost and constraint functions, and compute the sensitivities. These techniques are demonstrated in the crystal growth application by determining an optimal furnace wall temperature distribution to obtain the desired temperature profile in the crystal, and hence to maximize the crystal's quality. Several numerical optimization algorithms are studied to determine the proper convergence criteria, effective 1-D search strategies, appropriate forms of the cost and constraint functions, etc. In particular, we incorporate the conjugate gradient and quasi-Newton methods for unconstrained problems. The efficiency and effectiveness of each algorithm is presented in the example problem

Topology Optimization for the Full-Cell Design of Porous Electrodes in Electrochemical Energy Storage Devices

In this paper, we introduce a density-based topology optimization framework to design porous electrodes for maximum energy storage. We simulate the full cell with a model that incorporates electronic potential, ionic potential, and electrolyte concentration. The system consists of three materials, namely pure liquid electrolyte and the porous solids of the anode and cathode, for which we determine the optimal placement. We use separate electronic potentials to model each electrode, which allow interdigitated designs. As the result, a penalization is required to ensure that the anode and cathode do not touch, i.e. causing a short circuit. We compare multiple 2D designs generated for different fixed conditions, e.g. material properties. A 3D design with complex channel and interlocking structure is also created. All optimized designs are far superior to the traditional monolithic electrode design with respect to energy storage metrics. We observe up to 750% increase in energy storage for cases with slow effective ionic diffusion within the porous electrode

On internal constraints in continuum mechanics

When a body is subject to simple internal constraints, the deformation gradient must belong to a certain manifold. This is in contrast to the situation in the unconstrained case, where the deformation gradient is in an element of the open subset of second-order tensors with positive determinant. Commonly, following Truesdell and Noll, modern treatments of constrained theories start with an a priori additive decomposition of the stress into reactive and active components with the reactive component assumed to be powerless in all motions that satisfy the constraints and the active component given by a constitutive equation. Here, we obtain this same decomposition automatically by making a purely geometrical and general direct sum decomposition of the space of all second-order tensors in terms of the normal and tangent spaces of the constraint manifold. As an example, our approach is used to recover the familiar theory of constrained hyperelasticity.published or submitted for publicationis peer reviewe

Design sensitivity analysis for nonlinear dynamic thermoelastic systems

Two adjoint formulations are developed for the design sensitivity analysis of linear and nonlinear dynamic thermoelastic systems. The variation of a general design functional is expressed in explicit form with respect to variations of the design quantities, i.e. material properties, applied loads, prescribed boundary conditions, initial conditions, and shape. The design functional is expressed in terms of these explicit quantities and the implicit response variables: displacement, temperature, stress, strain, heat flux vector, temperature gradient, reaction forces, and reaction surface flux. The convolution is employed, in lieu of the time mappings used in other approaches to account for transient response. The design sensitivities for dynamic thermoelastic, nonlinear uncoupled dynamic thermoelastic problems are presented here for the first time. The use of the convolution for sensitivity analysis is also presented here for the first time. By formulating these sensitivities, numerical optimization algorithms currently used for elastostatic structure redesign, can be applied to nonlinear dynamic thermoelastic systems.Sensitivities for linear elastodynamic and dynamic thermoelastic problems are derived by incorporating the reciprocal theorem between a variation of the real design and an adjoint system. The Lagrange multiplier method is utilized to formulate sensitivities for nonlinear uncoupled dynamic thermoelastic problems. These formulations may also be specialized for linear and nonlinear transient conduction problems.The finite element method is used to demonstrate the numerical implementation of the formulations. Shape sensitivities are evaluated and compared with finite difference calculations to validate the results. A numerical optimization algorithm is combined with the sensitivity analyses to redesign the geometry of a mold in a casting problem and the shape of structure whose response is governed by a nonlinear quasi-static thermoelastic system.U of I OnlyETDs are only available to UIUC Users without author permissio

Constitutive Parameters and Their Evolution

The three-dimensional problem of anisotropic evolution is formulated and solved using the tools from optimal design. Basic assumptions are the principle of maximum energy dissipation and a function that relates the damage, as measured by the rate of a constitutive matrix norm, to an effective stress measure. The presentation relies on alternative description for the 21 constitutive parameters and only restrics the constitutive matrix to be semipositive definite