An adjoint method for the inverse design of solidification processes with natural convection

This paper presents a finite element algorithm based on the adjoint method for the design of a certain class of solidification processes. In particular, the paper addresses the design of directional solidification processes for pure materials such that a desired freezing front heat flux and growth velocity are achieved. This is the first time that an infinite-dimensional continuum adjoint formulation is obtained and implemented for the solution of such inverse/design problems with moving boundaries and Boussinesq incompressible flow. The present design problem belongs to a category of inverse problems in which one is looking for the unknown conditions in part of the boundary, while overspecified boundary conditions are supplied in another part of the boundary (here the freezing interface). The solidification design problem is mathematically posed as a whole time-domain optimization problem. The gradient of the cost functional is calculated using the solution of an appropriately defined continuous adjoint problem. The minimization process is realized by the conjugate gradient method via the solutions of the direct, adjoint and sensitivity sub-problems. The proposed methodology is demonstrated with the solidification of an initially superheated liquid aluminum confined in a square mold. The non-uniformity in the casting product in the direction of gravity due to the existence of natural convection in the melt is emphasized. The inverse design problem is then posed as finding the appropriate spatial-temporal variations of the boundary heat flux on the vertical mold walls that can eliminate or reduce the effects of convection on the freezing interface heat fluxes and growth velocity. The numerical example demonstrates the accuracy and convergence of the adjoint formulation. Finally, open related research design problems are discussed. © 1998 John Wiley & Sons, Ltd.     

Inverse design of directional solidification processes in the presence of a strong external magnetic field

A computational method for the design of directional alloy solidification processes is addressed such that a desired growth velocity ν_f under stable growth conditions is achieved. An externally imposed magnetic field is introduced to facilitate the design process and to reduce macrosegregation by the damping of melt flow. The design problem is posed as a functional optimization problem. The unknowns of the design problem are the thermal boundary conditions. The cost functional is taken as the square of the L₂ norm of an expression representing the deviation of the freezing interface thermal conditions from the conditions corresponding to local thermodynamic equilibrium. The adjoint method for the inverse design of continuum processes is adopted in this work. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, velocity and electric potential fields such that the gradient of the L₂ cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the FE solutions of the continuum direct, sensitivity and adjoint problems. The developed formulation is demonstrated with an example of designing the boundary thermal fluxes for the directional growth of a germanium melt with dopant impurities in the presence of an externally applied magnetic field. The design is shown to achieve a stable interface growth at a prescribed desired growth rate. Copyright © 2001 John Wiley & Sons, Ltd.     

Inverse problem of controlling the interface velocity in Stefan problems by conjugate gradient method

Abstract

The conjugate gradient method of minimization with adjoint equation is used successfully to solve the inverse problem in estimating an appropriate boundary control function such that the phase front moves at a desired velocity in the Stefan problem.

It is assumed that no prior information is available on the functional form of the unknown control function, therefore, it is classified as the function estimation in inverse calculation. The stability and accuracy of the inverse analysis using present algorithm are examined by comparing the results of the previous work by Voller [12].

Results show that the estimated control function by using conjugate gradient method did not exhibit oscillatory behavior in the inverse calculations for a broad range of front velocity while in [12] the inverse solutions are very sensitive to phase front velocity, therefore the application of future time stepping [2] is necessary in [12].

The advantage of applying this algorithm in inverse analysis lies in its stability as compared to the conventional minimization process [12]. Artificial future time stepping is unnecessary during the inverse calculation, since it is still an uncertainty in the inverse analysis. Furthermore, the inverse solutions obtained by the present method are found to be more accurate than the solutions obtained by the conventional minimization process.     

Control of the freezing interface morphology in solidification processes in the presence of natural convection

This paper presents a methodology for the solution of an inverse solidification design problem in the presence of natural convection. In particular, the boundary heat flux q₀ in the fixed mold wall, δΩ₀, is calculated such that a desired freezing front velocity and shape are obtained. As the front velocity together with the flux history q_ms on the solid side of the freezing front play a determinant role in the obtained cast structure, the potential applications of the proposed methods to the control of casting processes are enormous. The proposed technique consists of first solving a direct natural convection problem of the liquid phase in an a priori known shrinking cavity, Ω_L(t), before solving an ill-posed inverse design conduction problem in the solid phase in an a priori known growing region, Ω_S(t). The direct convection problem is used to evaluate the flux q_ml in the liquid side of the freezing front. A front tracking deforming finite element technique is employed. The flux q_ml can be used together with the Stefan condition to provide the freezing interface flux q_ms in the solid side of the front. As such, two boundary conditions (flux q_ms and freezing temperature θ_m) are especified along the (known) freezing interface δΩ_I(t). The developed design technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ∥θ_m – θ( x , t; q₀)∥, that expresses the square error between the calculated temperature θ( x , t; q₀) in the solid phase along δΩ_I(t) and the given melting temperature. The minimization of this cost functional is performed by the conjugate gradient method via the solutions of the direct, sensitivity and adjoint problems. A front tracking finite element technique is employed in this inverse analysis. Finally, an example is presented for the solidification of a superheated incompressible liquid aluminium, where the effects of natural convection in the moving interface shape are controlled with a proper adjustment of the cooling boundary conditions.     

On the solution of an ill-posed design solidification problem using minimization techniques in finite- and infinite-dimensional function spaces

This paper provides a comparative study of two alternative methodologies for the solution of an inverse design solidification problem. It is the one-dimensional solidification problem of calculating the boundary heat flux history that achieves a desired freezing front velocity and desired heat fluxes at the freezing front. The front velocity h(t) and flux history q_mS(t) on the solid side of the front control the obtained cast structure. As such, the potential applications of the proposed methods to the control of casting processes are enormous. The first technique utilizes a finite-dimensional approximation of the unknown boundary heat flux function q₀(t). The second technique uses the adjoint method to calculate in L₂ the derivative of the cost functional, ‖T_m – T(h(t), t;q₀)‖, that expresses the square error between the calculated T(h(t), t; q₀) and the given freezing front temperature T_m. Both steepest descent (SDM) and conjugate gradient methods (CGM) are examined. A front tracking FEM technique is used for the discretization of the state space. A detailed numerical analysis of the space and time discretization of the ‘parameter’ and state spaces, of the effect of the end condition of the adjoint problem and of other parameters in the solution are examined.     

Simultaneous reconstruction of the time-dependent Robin coefficient and heat flux in heat conduction problems

This paper aims to solve an inverse heat conduction problem in two-dimensional space under transient regime, which consists of the estimation of multiple time-dependent heat sources placed at the boundaries. Robin boundary condition (third type boundary condition) is considered at the working domain boundary. The simultaneous identification problem is formulated as a constrained minimization problem using the output least squares method with Tikhonov regularization. The properties of the continuous and discrete optimization problem are studied. Differentiability results and the adjoint problems are established. The numerical estimation is investigated using a modified conjugate gradient method. Furthermore, to verify the performance of the proposed algorithm, obtained results are compared with results obtained from the well-known finite-element software COMSOL Multiphysics under the same conditions. The numerical results show that the proposed algorithm is accurate, robust and capable of simultaneously representing the time effects on reconstructing the time-dependent Robin coefficient and heat flux.     

On the solution of an inverse natural convection problem using various conjugate gradient methods

The inverse problem of determining the time‐varying strength of a heat source, which causes natural convection in a two‐dimensional cavity, is considered. The Boussinesq equation is used to model the natural convection induced by the heat source. The inverse natural convection problem is solved through the minimization of a performance function utilizing the conjugate gradient method. The gradient of the performance function needed in the minimization procedure of the conjugate gradient method is obtained by employing either the adjoint variable method or the direct differentiation method. The accuracy and efficiency of these two methods are compared, and a new method is suggested that exploits the advantageous aspects of both methods while avoiding the shortcomings of them. Copyright © 2000 John Wiley & Sons, Ltd.     

Solution to the topology optimization problem using a time-evolution equation

This article describes a numerical solution to the topology optimization problem using a time-evolution equation. The design variables of the topology optimization problem are defined as a mathematical scalar function in a given design domain. The scalar function is projected to the normalized density function. The adjoint variable method is used to determine the gradient defined as the ratio of the variation of the objective function or constraint function to the variation of the design variable. The variation of design variables is obtained using the solution of the time-evolution equation in which the source term and Neumann boundary condition are given as a negative gradient. The distribution of design variables yielding an optimal solution is obtained by time integration of the solution of the time-evolution equation. By solving the topology optimization problem using the proposed method, it is shown that the objective function decreases when the constraints are satisfied. Furthermore, we apply the proposed method to the thermal resistance minimization problem under the total volume constraint and the mean compliance minimization problem under the total volume constraint.     

Identification of boundary displacements in plane elasticity by BEM

The purpose of this study is to present a possible application of BEM for numerical identification of the boundary conditions for Navier equations in plane elasticity with internal measurements, based on insufficient and noisy information for unique identification. The inverse problem is re-formulated as a minimization problem by the direct variational method. The minimization problem is then recast using the gradient method into successive primary and adjoint boundary value problems in the corresponding plane elasticity problem. For numerical solution of the elasticity problems, the conventional direct boundary element method is employed. From the simple numerical examples considered, it is concluded that our identification scheme is stable and the approximate solutions are convergent to the minimum.     

A cell-by-cell thermally driven mushy cell tracking algorithm for phase-change problems

A thermally driven mushy cell tracking algorithm for phase-change problems with a moving boundary is presented. The equation used to track the moving boundary is based on energy balance over the mushy cell and is applied to advance a moving front in a cell-by-cell manner. The efficacy of the tracking algorithm is demonstrated on specific problems solved using the finite volume method. An implicit scheme is adopted to ensure that the numerical solution is unconditionally stable in time. A preconditioned conjugated gradient (P-CG) solver is implemented to ensure that solutions converge in a finite number of iterations. Four benchmark cases are used to validate the algorithm including solidification in one dimensional space (two-region problem), melting of pure aluminum in two-dimensional (2D) space, solidification with periodic boundary conditions, and solidification of one-region problem. The results obtained show that the current algorithm is capable of converging to accurate solutions for moving fronts and the numerical predications are in excellent agreement with corresponding analytical solutions.     

Control of Unsteady Solidification Via Optimized Magnetic Fields

This article presents a numerical procedure for automatically controlling desired features of a melt undergoing solidification by applying an external magnetic field whose time-varying intensity and spatial distribution are obtained by the use of a hybrid optimization algorithm. The intensities of the magnets along the boundaries of the container were discretized by using B-splines. The inverse problem is then formulated to find the magnetic boundary conditions (the coefficients of the B-splines) in such a way that the gradients of temperature along the gravity direction are minimized at each instant as the solidification front advances through a moving melt. For this task, a hybrid optimization code was used that automatically switches among the following six optimization modules; the Davidon-Fletcher-Powell (DFP) gradient method, a genetic algorithm (GA), the Nelder-Mead (NM) simplex method, quasi-Newton algorithm of Pshenichny-Danilin (LM), differential evolution (DE), and sequential quadratic programming (SQP). Transient Navier-Stokes and Maxwell's equations were discretized by using a finite volume method in a generalized curvilinear nonorthogonal coordinate system. For the phase change problems, an enthalpy formulation was used. The computer code was validated against analytical and numerical benchmark results with very good agreements in both cases.     

Two-dimensional inverse profiling problem using phaseless data

We discuss the characterization of two-dimensional targets based on their diffracted intensity. The target characterization is performed by minimizing an adequate cost functional, combined with a level-set representation if the target is homogeneous. One key issue in this minimization is the choice of an updating direction, which involves the gradient of the cost functional. This gradient can be evaluated using a fictitious field, the solution of an adjoint problem in which receivers act as sources with a specific amplitude. We explore the Born approximation for the adjoint field and compare various approaches for a wide variety of objects.     

Mathematical simulation of metal solidification in a wedge-like casting mold with allowance for natural convection

Using the finite difference method, a nonstationary problem of metal solidification in a wedge-like casting mold has been solved in a two-dimensional statement with allowance for natural convection. For isolated instants of time the positions of the solidification front, the profiles of temperature, horizontal and vertical velocity components, and of the stream function have been obtained.     

Shape optimization using time evolution equations

In this study, we deal with a numerical solution based on time evolution equations to solve the optimization problem for finding the shape that minimizes the objective function under given constraints. The design variables of the shape optimization problem are defined on a given original domain on which the boundary value problems of partial differential equations are defined. The variations of the domain are obtained by the time integration of the solution to derive the time evolution equations defined in the original domain. The shape gradient with respect to the domain variations are given as the Neumann boundary condition defined on the original domain boundary. When the constraints are satisfied, the decreasing property of the objective function is guaranteed by the proposed method. Furthermore, the proposed method is used to minimize the heat resistance under a total volume constraint and to solve the minimization problem of mean compliance under a total volume constraint.     

Gradient method for inverse heat conduction problem in nanoscale

An inverse heat conduction problem for nanoscale structures was studied. The conduction phenomenon is modelled using the Boltzmann transport equation. Phonon‐mediated heat conduction in one dimension is considered. One boundary, where temperature observation takes place, is subject to a known boundary condition and the other boundary is exposed to an unknown temperature. The gradient method is employed to solve the described inverse problem. The sensitivity, adjoint and gradient equations are derived. Sample results are presented and discussed. Copyright © 2004 John Wiley & Sons, Ltd.     

Heat transfer with solid liquid phase change initiating at a point in a semi infinite mold

Short-time analytical solutions of temperature and moving boundary in two-dimensional two-phase freezing due to a cold spot are presented in this paper. The melt occupies a semi-infinite region. Although the method of solution is valid for various other types of boundary conditions, the results in this paper are given only for the prescribed flux boundary conditions which could be space and time dependent. The freezing front propagations along the interior of the melt region exhibit well known behaviours but the propagations along the surface are of new type. The freezing front always depends on material parameters. Several interesting results can be obtained as particular cases of the general results.     

Second-order adjoint equation method for parameter identification of rock based on blast waves in tunnel excavation

The objective of this study is to present a method for identifying the elastic moduli of ground rock via the first- and second-order adjoint methods using blast vibration measurements during tunnel excavation. For identifying these parameters, the magnitudes of the blast force should be identified beforehand. Parameter identification is a minimization problem of the square sum of the discrepancy between the computed and observed velocities. The magnitudes of the three components of borehole pressure are assumed to be independent in each direction. The propagation of an elastic wave is assumed because the amplitude of such a wave is infinitesimal. The three-dimensional finite element method and the linear acceleration method are used effectively. The extended performance function can be expanded into a series of small constants to derive the necessary condition of minimization. The adjoint equation and the dynamic equation of motion can be used to obtain the gradient and the Hessian product of the extended performance function with respect to the parameters. The weighted gradient method and Broyden–Flecher–Goldfarb–Shanno method are successfully employed for the minimization. By applying the present identification technique at the Ohyorogi tunnel site, the fact that the computed and observed velocities are well in agreement is verified. The present method can be shown to be useful for tunnel excavation.     

AN INVERSE PROBLEM IN PREDICTING TEMPERATURE DEPENDENT HEAT CAPACITY PER UNIT VOLUME WITHOUT INTERNAL MEASUREMENTS

An inverse analysis utilizing the conjugate gradient method and the minimization of adjoint equation is used successfully to estimate the temperature dependent heat capacity per unit volume in a homogeneous medium. No prior information is available on the functional form of the unknown heat capacity in the present study, thus, it is classified as the function estimation in inverse calculation. The accuracy of the inverse analysis is examined by using simulated exact and inexact measurements obtained within the medium. Results show that an excellent estimation on the heat capacity per unit volume can be obtained by using just boundary measurements (i.e. internal measurements are unnecessary) and the CPU time used in VAX-9420 computer is less than one second. The advantages of applying this algorithm in inverse analysis can greatly simplify the experimental set-up, diminish the sensitivity to the measurement errors and reduce the CPU time in inverse calculation, while the reliable predictions can still be achieved.     

An application of the symplectic system in two-dimensional viscoelasticity

This paper redescribes fundamental problem of the two-dimensional viscoelasticity in symplectic system. With the aid of the symplectic character and integral transformation, solutions of duality equations are obtained, or Saint-Venant solutions of extension and bend and local solutions of boundary effects. Thus the original problem is reduced to finding zero eigenvalue eigensolutions and non-zero eigenvalue eigensolutions. Meanwhile, adjoint relationships of the symplectic orthogonality in the Laplace domain are generalized to in the time domain. After obtaining fundamental eigensolutions, the problem can be discussed in the eigensolution space of the time domain without the need of the Laplace transformation and inverse one. As its application, a direct method is shown and some examples are discussed, which reveal relations between the creep or relaxation and eigensolutions. The symplectic method and numerical method provide an idea for other researching as well.     

The optimal control of unsteady flows with a discrete adjoint method

This paper presents a general framework to derive a discrete adjoint method for the optimal control of unsteady flows. The complete formulation of a generic time-dependent optimal design problem is introduced and it is outlined how to derive the discrete set of adjoint equations in a general approach. Results are shown that demonstrate the application of the theory to the drag minimization of viscous flow around a rotating cylinder, and to the remote inverse design of laminar flow around the multi-element NLR 7301 configuration at a high angle of attack. In order to reduce the considerable computational costs of unsteady optimization, the use of bigger time steps over transitional or unphysical adjusting periods as well as omitting time steps while recording the flow solution are investigated and are shown to work well in practice.