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.     

An inverse problem in estimating the relaxation time for nanoscale phonon radiative transfer problem

An inverse nanoscale phonon radiative transfer problem is solved in this study by using conjugate gradient method (CGM) to estimate the unknown frequency‐ and temperature‐dependent relaxation time, based on the simulated phonon intensity measurements. The CGM in dealing with the present integro‐differential governing equations is not as straightforward as for the normal differential equations; special treatments are needed to overcome the difficulties. Results obtained in this inverse analysis will be justified based on the numerical experiments where two different unknown distributions of relaxation time are to be estimated. Finally, it is shown that the reliable frequency and temperature‐dependent relaxation time can be obtained with CGM. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

A boundary element-based inverse-problem in estimating transient boundary conditions with conjugate gradient method

A Boundary Element Method (BEM)-based inverse algorithm utilizing the iterative regularization method, i.e. the conjugate gradient method (CGM), is used to solve the Inverse Heat Conduction Problem (IHCP) of estimating the unknown transient boundary temperatures in a multi-dimensional domain with arbitrary geometry. The results obtained by the CGM are compared with that obtained by the standard Regularization Method (RM). The error estimation based on the statistical analysis is derived from the formulation of the RM. A 99 per cent confidence bound is thus obtained. Finally, the effects of the measurement errors to the inverse solutions are discussed. Results show that the advantages of applying the CGM in the inverse calculations lie in that (i) the major difficulties in choosing a suitable quadratic norm, determining a proper regularization order and determining the optimal smoothing (or regularization) coefficient in the RM are avoided and (ii) it is less sensitive to the measurement errors, i.e. more accurate solutions are obtained. © 1998 John Wiley & Sons, Ltd.     

Numerical method for the solution of non‐linear two‐dimensional inverse heat conduction problem using unstructured meshes

A new method of solving multidimensional heat conduction problems is formulated. The developed space marching method allows to determine quickly and exactly unsteady temperature distributions in the construction elements of irregular geometry. The method which is based on temperature measurements at the outer surface, is especially appropriate for determining transient temperature distribution in thick‐wall pressure components. Two examples are included to demonstrate the capabilities of the new approach. Copyright © 2000 John Wiley & Sons, Ltd.     

Solution of the time‐harmonic viscoelastic inverse problem with interior data in two dimensions

We consider the problem of determining the distribution of the complex‐valued shear modulus for an incompressible linear viscoelastic material undergoing infinitesimal time‐harmonic deformation, given the knowledge of the displacement field in its interior. In particular, we focus on the two‐dimensional problems of anti‐plane shear and plane stress. These problems are motivated by applications in biomechanical imaging, where the material modulus distributions are used to detect and/or diagnose cancerous tumors. We analyze the well‐posedness of the strong form of these problems and conclude that for the solution to exist, the measured displacement field is required to satisfy rather restrictive compatibility conditions. We propose a weak, or a variational formulation, and prove the existence and uniqueness of solutions under milder conditions on measured data. This formulation is derived by weighting the original PDE for the shear modulus by the adjoint operator acting on the complex‐conjugate of the weighting functions. For this reason, we refer to it as the complex adjoint weighted equation (CAWE). We consider a straightforward finite element discretization of these equations with total variation regularization, and test its performance with synthetically generated and experimentally measured data. We find that the CAWE method is, in general, less diffusive than a corresponding least squares solution, and that the total variation regularization significantly improves its performance in the presence of noise. Copyright © 2012 John Wiley & Sons, Ltd.     

Conjugate gradient method for the Robin inverse problem associated with the Laplace equation

This paper studies a non-linear inverse problem associated with the Laplace equation of identifying the Robin coefficient from boundary measurements. A variational formulation of the problem is suggested, thereby transforming it into an optimization problem. Mathematical properties relevant to its numerical computation are established. The optimization problem is solved using the conjugate gradient method in conjunction with the discrepancy principle, and the algorithm is implemented using the boundary element method. Numerical results are presented for several benchmark problems with both exact and noisy data, and the convergence of the algorithm with respect to mesh refinement and decreasing the amount of noise in the data is investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

The identification of a Robin coefficient by a conjugate gradient method

This paper investigates a non‐linear inverse problem associated with the heat conduction problem of identifying a Robin coefficient from boundary temperature measurement. The variational formulation of the problem is given. The conjugate gradient method combining with the discrepancy principle for choosing the suitable stop step are proposed for solving the optimization problem, in which the finite difference method is used to solve the direct problems. The performance of the method is verified by simulating four examples. The convergence with respect to the grid refinement and the amount of noise in the data is also investigated. Copyright © 2008 John Wiley & Sons, Ltd.     

A Fourier‐series‐based virtual fields method for the identification of 2‐D stiffness distributions

The virtual fields method (VFM) is a powerful technique for the calculation of spatial distributions of material properties from experimentally determined displacement fields. A Fourier‐series‐based extension to the VFM (the F‐VFM) is presented here, in which the unknown stiffness distribution is parameterised in the spatial frequency domain rather than in the spatial domain as used in the classical VFM. We present in this paper the theory of the F‐VFM for the case of elastic isotropic thin structures with known boundary conditions. An efficient numerical algorithm based on the two‐dimensional Fast Fourier Transform (FFT) is presented, which reduces the computation time by three to four orders of magnitude compared with a direct implementation of the F‐VFM for typical experimental dataset sizes. Artefacts specific to the F‐VFM (ringing at the highest spatial frequency near to modulus discontinuities) can be largely removed through the use of appropriate filtering strategies. Reconstruction of stiffness distributions with the F‐VFM has been validated on three stiffness distribution scenarios under varying levels of noise in the input displacement fields. Robust reconstructions are achieved even when the displacement noise is higher than in typical experimental fields.Copyright © 2014 John Wiley & Sons, Ltd.     

A computational method for inverse free boundary determination problem

Based on the method of fundamental solutions and discrepancy principle for the choice of location for source points, we extend in this paper the application of the computational method to determine an unknown free boundary of a Cauchy problem of parabolic‐type equation from measured Dirichlet and Neumann data with noises. The standard Tikhonov regularization technique with the L‐curve method for an optimal regularized parameter is adopted for solving the resultant highly ill‐conditioned system of linear equations. Both one‐dimensional and two‐dimensional numerical examples are given to verify the efficiency and accuracy of the proposed computational method. Copyright © 2007 John Wiley & Sons, Ltd.     

Adjoint‐weighted variational formulation for the direct solution of inverse problems of general linear elasticity with full interior data

We describe a novel variational formulation of inverse elasticity problems given interior data. The class of problems considered is rather general and includes, as special cases, plane deformations, compressibility and incompressiblity in isotropic materials, 3D deformations, and anisotropy. The strong form of this problem is governed by equations of pure advective transport. The variational formulation is based on a generalization of the adjoint‐weighted variational equation (AWE) formulation, originally developed for flow of a passive scalar. We describe how to apply AWE to various cases, and prove several properties. We prove that the Galerkin discretization of the AWE formulation leads to a stable, convergent numerical method, and prove optimal rates of convergence. The numerical examples demonstrate optimal convergence of the method with mesh refinement for multiple unknown material parameters, graceful performance in the presence of noise, and robust behavior of the method when the target solution is C^∞, C⁰, or discontinuous. Copyright © 2009 John Wiley & Sons, Ltd.     

Dynamic stiffness vibration analysis using a high‐order beam model

This work presents the derivation of the exact dynamic stiffness matrix for a high‐order beam element. The terms are found directly from the solutions of the differential equations that describe the deformations of the cross‐section according to the high‐order theory, which include cubic variation of the axial displacements over the cross‐section of the beam. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments. Using the dynamic stiffness matrix exact vibration frequencies for beams with various combinations of boundary conditions are tabulated and compared with results from the Bernoulli–Euler and Timoshenko beam models. Copyright © 2003 John Wiley & Sons, Ltd.     

A sensitivity-based approach to solving the inverse eigenvalue problem for linear structures carrying lumped attachments

This paper presents an efficient algorithm for designing dynamical systems to exhibit a desired spectrum of eigenvalues. Focusing on combined systems of linear structures carrying various lumped element attachments, we apply the assumed-modes method and the implicit function theorem to derive analytical expressions for eigenvalue sensitivities, which are used to efficiently determine the minimal set of structural modifications needed to achieve a set of desired eigenvalues. The proposed algorithm employs an adaptive step size, performs significantly better than existing approaches, and can be easily applied to a broad range of structures. Convergence properties and limitations on achievable eigenvalues are also discussed, and a number of case studies demonstrating the performance of the algorithm in a wide variety of different applications are also included.     

A non‐iterative finite element method for inverse heat conduction problems

A non‐iterative, finite element‐based inverse method for estimating surface heat flux histories on thermally conducting bodies is developed. The technique, which accommodates both linear and non‐linear problems, and which sequentially minimizes the least squares error norm between corresponding sets of measured and computed temperatures, takes advantage of the linearity between computed temperatures and the instantaneous surface heat flux distribution. Explicit minimization of the instantaneous error norm thus leads to a linear system, i.e. a matrix normal equation, in the current set of nodal surface fluxes. The technique is first validated against a simple analytical quenching model. Simulated low‐noise measurements, generated using the analytical model, lead to heat transfer coefficient estimates that are within 1% of actual values. Simulated high‐noise measurements lead to h estimates that oscillate about the low‐noise solution. Extensions of the present method, designed to smooth oscillatory solutions, and based on future time steps or regularization, are briefly described. The method's ability to resolve highly transient, early‐time heat transfer is also examined; it is found that time resolution decreases linearly with distance to the nearest subsurface measurement site. Once validated, the technique is used to investigate surface heat transfer during experimental quenching of cylinders. Comparison with an earlier inverse analysis of a similar experiment shows that the present method provides solutions that are fully consistent with the earlier results. Although the technique is illustrated using a simple one‐dimensional example, the method can be readily extended to multidimensional problems. Copyright © 2003 John Wiley & Sons, Ltd.     

A time‐parallel implicit method for accelerating the solution of non‐linear structural dynamics problems

The parallel implicit time‐integration algorithm (PITA) is among a very limited number of time‐integrators that have been successfully applied to the time‐parallel solution of linear second‐order hyperbolic problems such as those encountered in structural dynamics. Time‐parallelism can be of paramount importance to fast computations, for example, when space‐parallelism is unfeasible as in problems with a relatively small number of degrees of freedom in general, and reduced‐order model applications in particular, or when reaching the fastest possible CPU time is desired and requires the exploitation of both space‐ and time‐parallelisms. This paper extends the previously developed PITA to the non‐linear case. It also demonstrates its application to the reduction of the time‐to‐solution on a Linux cluster of sample non‐linear structural dynamics problems. Copyright © 2008 John Wiley & Sons, Ltd.     

A fast collocation method for an inverse boundary value problem

In this paper, we present an implementation of a fast multiscale collocation method for boundary integral equations of the second kind, and its application to solving an inverse boundary value problem of recovering a coefficient function from a boundary measurement. We illustrate by numerical examples the insensitive nature of the map from the coefficient to measurement, and design and test a Gauss–Newton iteration algorithm for obtaining the best estimate of the unknown coefficient from the given measurement based on a least‐squares formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

An iterative regularization method in estimating the unknown energy source by laser pulses with a dual-phase-lag model

In the present study an inverse problem for hyperbolic heat conduction with a dual-phase-lag model is solved by the conjugate gradient method (CGM) in estimating the unknown heat generation, due to the ultra-short duration laser heating, based on the interior temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where two different heat source distributions are to be estimated. Results show that the inverse solutions can always be obtained when choosing the initial guesses of the heat sources equal to zero. Finally, it is concluded that accurate heat sources can be estimated in this study. Copyright © 2008 John Wiley & Sons, Ltd.     

An optimal control problem in estimating the optimal control force for the force vibration system

An optimal control problem for the force vibration system based on the iterative regularization method, i.e. the conjugate gradient method (CGM), is examined to estimate the optimal control force in a damped system having time‐dependent system parameters such that the desire (or design) system displacements can be satisfied. It is assumed that no prior information is available on the functional form of the unknown control function in the present study, thus, it is classified as the function estimation. Numerical simulations are performed to test the validity of the present algorithm by using different types of the desire system displacements. Results show that an excellent estimation on the optimal control force can be obtained with arbitrary initial guesses within a couple of second's CPU time at Pentium III‐500 MHz PC. Copyright © 2001 John Wiley & Sons, Ltd.     

Multi‐time‐step explicit–implicit method for non‐linear structural dynamics

We present a method with domain decomposition to solve time‐dependent non‐linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non‐linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non‐linear situations in the following cases: implicit linear/explicit non‐linear, explicit non‐linear/explicit non‐linear and implicit non‐linear/explicit non‐linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.