Direct error in constitutive equation formulation for inverse heat conduction problem

We present a mixed numerical formulation that handles discontinuities well for scalar hyperbolic partial differential equations. The formulation is based on a least‐square error in the constitutive equation. It is motivated by scalar inverse diffusion problems with interior data and applies to convection of a passive scalar in a discontinuous compressible flow field. We motivate the need for a mixed formulation by discretizing using an irreducible finite element method and discuss some of the limitations of that approach. We then develop and prove that the mixed formulation is well posed and verify that it works for problems with continuous and discontinuous thermal conductivity distributions.     

Inverse material identification in coupled acoustic-structure interaction using a modified error in constitutive equation functional

This work focuses on the identification of heterogeneous linear elastic moduli in the context of frequency-domain, coupled acoustic-structure interaction (ASI), using either solid displacement or fluid pressure measurement data. The approach postulates the inverse problem as an optimization problem where the solution is obtained by minimizing a modified error in constitutive equation (MECE) functional. The latter measures the discrepancy in the constitutive equations that connect kinematically admissible strains and dynamically admissible stresses, while incorporating the measurement data as additional quadratic error terms. We demonstrate two strategies for selecting the MECE weighting coefficient to produce regularized solutions to the ill-posed identification problem: 1) the discrepancy principle of Morozov, and 2) an error-balance approach that selects the weight parameter as the minimizer of another functional involving the ECE and the data misfit. Numerical results demonstrate that the proposed methodology can successfully recover elastic parameters in 2D and 3D ASI systems from response measurements taken in either the solid or fluid subdomains. Furthermore, both regularization strategies are shown to produce accurate reconstructions when the measurement data is polluted with noise. The discrepancy principle is shown to produce nearly optimal solutions, while the error-balance approach, although not optimal, remains effective and does not need a priori information on the noise level.     

A 3D domain decomposition approach for the identification of spatially varying elastic material parameters

The post‐treatment of (3D) displacement fields for the identification of spatially varying elastic material parameters is a large inverse problem that remains out of reach for massive 3D structures. We explore here the potential of the constitutive compatibility method for tackling such an inverse problem, provided an appropriate domain decomposition technique is introduced. In the method described here, the statically admissible stress field that can be related through the known constitutive symmetry to the kinematic observations is sought through minimization of an objective function, which measures the violation of constitutive compatibility. After this stress reconstruction, the local material parameters are identified with the given kinematic observations using the constitutive equation. Here, we first adapt this method to solve 3D identification problems and then implement it within a domain decomposition framework which allows for reduced computational load when handling larger problems. Copyright © 2015 John Wiley & Sons, Ltd.     

New approach to identify the initial condition in degenerate hyperbolic equation

This paper deals with the determination of an initial condition in degenerate hyperbolic equation from final observations. With the aim of reducing the execution time, this inverse problem is solved using an approach based on double regularization: a Tikhonov’s regularization and regularization in equation by viscose-elasticity. So, we obtain a sequence of weak solutions of degenerate linear viscose-elastic problems. Firstly, we prove the existence and uniqueness of each term of this sequence. Secondly, we prove the convergence of this sequence to the weak solution of the initial problem. Also we present some numerical experiments to show the performance of this approach.     

Inverse identification of constitutive parameters from heat source fields: A local approach applied to hyperelasticity

In this paper, a new inverse identification method of constitutive parameters is developed from full kinematic and thermal field measurements. It consists in reconstructing the heat source field from two different approaches by using the heat diffusion equation. The first one requires the temperature field measurement and the value of the thermophysical parameters. The second one is based on the kinematic field measurement and the choice of a thermo-hyperelastic model that contains the parameters to be identified. The identification is carried out at the local scale, ie, at any point of the heat source field, without using the boundary conditions. In the present work, the method is applied to the challenging case of hyperelasticity from a heterogeneous test. Due to large deformations undergone by the rubber specimen tested, a motion compensation technique is developed to plot the kinematic and the thermal fields at the same points before reconstructing the heterogeneous heat source field. In the present case, the constitutive parameter of the Neo-Hookean model has been identified, and its distribution has been characterized with respect to the strain state at the surface of a cross-shaped specimen.     

Coupling recognition of the structure and parameters of non‐linear constitutive material models using hybrid evolutionary algorithms

The structure of the non‐linear constitutive models is a key to control non‐linear behaviours of materials. Because the non‐linear mechanical mechanism is not clearly understood in most cases, it is very difficult to assume the structure of the model in advance. The recognition of the structure of the model from experimental results can help understanding of the mechanism. This recognition is a dynamic search problem being highly multimodal, multi‐variable with high order, and needing a large parameter space. How to obtain a global optimum solution is a key to this problem. In this paper, a hybrid evolutionary algorithm is proposed for coupling recognition of the structure of the non‐linear constitutive material model and its coefficients in global space using global response information, e.g. load vs deflection data, obtained from the structural test. Genetic programming is used to recognize the structure of the non‐linear stress–strain relationship without any assumption in advance and the genetic algorithm is then used to recognize its coefficients. The non‐linear stress–strain relationship thus found can not only satisfy the dynamic change in its structure but also its variables and coefficients. Non‐linear finite element analysis is used to transfer the load–deflection information to the stress–strain data. The potential of the proposed method is demonstrated by applying it to the macro‐mechanical modelling of the non‐linear behaviour of composite materials. A non‐linear material model for the unidirectional ply is recognized by using experimental data of a lamina plate [(±45)₆]_s. The obtained non‐linear constitutive model gave good predictions in coincidence with the non‐linear behaviours of the [(±30)₆]_s, [(0/±45)₃]_s and [(0/±45)₄]_s plates. The results indicate that the coupling non‐linear constitutive model of the structure and its coefficients can identify the model which the traditional constitutive model theory is unable to recognize. Copyright © 2004 John Wiley & Sons, Ltd.     

A Dissipation Gap Method for full‐field measurement‐based identification of elasto‐plastic material parameters

Using enriched data such as displacement fields obtained from digital image correlation is a pathway to the local identification of material parameters. Up to now, most of the identification techniques for nonlinear models are based on Finite Element Updating Methods. This article explains how an appropriate use of the Dissipation Gap Method can help in this context and be an interesting alternative to these classical techniques. The Dissipation Gap Methods rely on the concept of error in dissipation that has been used mainly for the verification of finite element simulations. We provide here an original application of these founding developments to the identification of material parameters for nonlinear behaviors. The proposed technique and especially the main technical keypoint of building the admissible fields are described in detail. The approach is then illustrated through the identification of heterogeneous isotropic elasto‐plastic properties. The basic numerical features highlighted through these simple examples demonstrate this approach to be a promising tool for nonlinear identification.Copyright © 2012 John Wiley & Sons, Ltd.     

Real‐time updating of structural mechanics models using Kalman filtering,modified constitutive relation error,and proper generalized decomposition

The paper aims at proposing a new strategy for real‐time identification or updating of structural mechanics models defined as dynamical systems. The main idea is to introduce the modified constitutive relation error concept, which is a practical tool that enables to efficiently solve identification problems with highly corrupted data, into the Kalman filtering, which is a classical framework for data assimilation. Furthermore, a PGD‐based model reduction method is performed in order to optimize capabilities of the online updating strategy. Performances of the proposed approach, in terms of robustness gain and computational cost reduction, are illustrated on several unsteady thermal applications. Copyright © 2015 John Wiley & Sons, Ltd.     

Constitutive equation for a class of isotropic, perfectly elastic solids using a new measure of finite strain and corresponding stress

The definition of a measure of strain, referred to as the bi-configuration strain tensor, centres on the difference between the left Cauchy-Green deformation tensor and its inverse. A new measure of stress, coined the bi-configuration stress tensor, has been defined. This measure of stress refers the traction in the current configuration jointly to the referential and spatial configurations, that is, to an effective element of area identified as an element of bi-configuration area. The stress and strain tensors are assumed to be constitutively related by a finite strain form of a generalised Hooke's law. The predictions obtained from the proposed constitutive equation are compared with the observed mechanical behaviour of various test materials. Comparison with experiment centres on biaxial stress measurements in various simple modes of deformation identified by way of a generalised stress-strain relation. The predictions from the proposed constitutive theory are in good accord with the results of experiment.     

Generalized stochastic approach for constitutive equation in linear elasticity: a random matrix model

This work is concerned with the construction of stochastic models for random elasticity matrices, allowing either for the generation of elasticity tensors exhibiting some material symmetry properties almost surely (integrating the statistical dependence between the random stiffness components) or for the modeling of random media that requires the mean of a stochastic anisotropy measure to be controlled apart from the level of statistical fluctuations. To this aim, we first introduced a decomposition of the stochastic elasticity tensor on a deterministic tensor basis and considered the probabilistic modeling of the random components, having recourse to the MaxEnt principle. Strategies for random generation and estimation were further reviewed, and the approach was exemplified in the case of a material that was transversely isotropic almost surely. In a second stage, we made use of such derivations to propose a generalized model for random elasticity matrices that took into account, almost separately, constraints on both the level of stochastic anisotropy and the level of statistical fluctuations. An example was finally provided and showed the efficiency of the approach. Copyright © 2011 John Wiley & Sons, Ltd.     

Damage identification in framed structures using natural frequencies

A novel procedure for damage identification of framed structures is proposed, where both the location and the extent of structural damage in framed structures can be correctly determined using only a limited number of measured natural frequencies. No knowledge of the modal shapes of the damaged structure is required. On the basis of the characteristic equations for the original and the damaged structure, a set of equations is generated. Two computational techniques, the direct iteration (DI) technique and the Gauss–Newton least-squares (GNLS) technique, are utilized to determine structural damage from the derived equations. Finally, different numerical examples are used to demonstrate the effectiveness of the proposed method. © 1997 John Wiley & Sons, Ltd.     

Detection of material interfaces using a regularized level set method in piezoelectric structures

An algorithm to solve the inverse problem of detecting inclusion interfaces in a piezoelectric structure is proposed. The material interfaces are implicitly represented by level sets which are identified by applying regularization using total variation penalty terms. The inverse problem is solved iteratively and the extended finite element method is used for the analysis of the structure in each iteration. The formulation is presented for three-dimensional structures and inclusions made of different materials are detected by using multiple level sets. The results obtained prove that the iterative procedure proposed can determine the location and approximate shape of material sub-domains in the presence of higher noise levels.     

Analysis of transfer procedures in elastoplasticity based on the error in the constitutive equations: Theory and numerical illustration

The aim of this work is to illustrate a methodology for the assessment of adaptive strategies for the solution of associative rate‐independent plasticity problems solved by employing the incremental displacement conforming finite element method. This is the first step towards a more rational definition of transfer operators in terms of the ensuing error. The motivating idea is the observation that change of data and/or finite element mesh from one time interval to the other can be both related to a discontinuity jump of the approximate solution across the time instant t_n. Thus, reliable a posteriori estimates will have to depend not only on the time step and finite element mesh size but also on the value of the jump. A new error estimate based on the error in the constitutive equations is developed which allows characterization of the discontinuity jump. Copyright © 2004 John Wiley & Sons, Ltd.     

Iterative computational identification of a space-wise dependent source in parabolic equation

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time – the final overdetermination. More general problems are associated with some integral observation of the solution in time – the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spatial variables. Capabilities of proposed methods are illustrated by numerical examples for a two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation in space is used, whilst the time discretization is based on fully implicit two-level schemes.     

Numerical inversions for diffusion coefficients in two-dimensional space fractional diffusion equation

This article deals with an inverse problem of determining the diffusion coefficients in 2D fractional diffusion equation with a Dirichlet boundary condition by the final observations at the final time. The forward problem is solved by the alternating direction implicit finite-difference scheme with the discrete of fractional derivative by shift Grünwald formula and a numerical text which is to prove its numerically stability and convergence is given. Furthermore, the homotopy regularization algorithm with the regularization parameter chosen by a Sigmoid-type function is introduced to solve the inversion problem numerically. Numerical inversions both with accurate data and noisy data are carried out for the unknown diffusion coefficients of constant and variable with polynomials, trigonometric and index functions. The reconstruction results show that the inversion algorithm is efficient for the inverse problem of determining diffusion coefficients in 2D space fractional diffusion equation, and the algorithm is also numerically stable for additional date having random noises.     

光学CT中扩散方程光子传输模型的Monte Carlo验证

扩散方程模型被广泛应用于光学CT图像的重建中,其精确性直接关系到所成图像的质量。为了研究扩散方程模型对近红外光子输运过程描述的精确性及其适用范围,本文应用Monte Carlo（MC）模拟方法,给出了二维组织体光学CT正向问题的数值模拟结果,并与其扩散方程模型的有限元解进行了分析比较。实验结果表明,基于扩散方程的光子传输模型能较为准确地模拟光子在强散射介质中的输运过程,适用于该种光学参数条件下的图像重建研究。     

On the plane problem approximation in the theory of macroisotropic composite material

A plane problem approximation is studied with mean, over the volume of phases, stresses in a two-phase macroisotropic composite material. Numerical results for WC-Co hard metals with a boundary type microstructure are analyzed. The microstresses as determined within the plane strain approach can be higher than the true stresses in the case of simple shearing strain but lower than the true stresses in the case of a uniaxial stressed state; the residual thermal stresses are smaller than the true residual stresses. The stresses estimated by the two-and three-dimensional approaches tend to equalize as the cobalt binder amount is raised. __________ Translated from Problemy Prochnosti, No. 1, pp. 13–21, January–February, 2007.     

Guaranteed energy error bounds for the Poisson equation using a flux‐free approach: Solving the local problems in subdomains

A method to compute guaranteed upper bounds for the energy norm of the exact error in the finite element solution of the Poisson equation is presented. The bounds are guaranteed for any finite element mesh however coarse it may be, not just in the asymptotic regime. The bounds are constructed by employing a subdomain‐based a posteriori error estimate which yields self‐equilibrated residual loads in stars (patches of elements). The proposed approach is an alternative to standard equilibrated residual methods providing sharper bounds. The use of a flux‐free error estimator improves the effectivities of the upper bounds for the energy while retaining the certainty of the bounds. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical solution of two-dimensional inverse force function in the wave equation with nonlocal boundary conditions

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse time-dependent force function in the wave equation on regular and irregular domains. The SMRPI is developed for identifying the force function which satisfies in the wave equation subject to the integral overspecification over a portion of the spatial domain or to the overspecification at a point in the spatial domain. This method is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. Since the problem is known to be ill-posed, Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system. Three numerical examples are tested to show that numerical results are accurate for exact data and stable with noisy data.     

A non-iterative method for identifying multiple unknown time-dependent sources compactly supported occurring in a 2D parabolic equation

We address the non-linear inverse source problem of identifying multiple unknown time-dependent sources spatially supported in some subdomains of a two-dimensional bounded domain subject to an evolution advection–dispersion–reaction equation. Provided to be available within the monitored domain interfaces for recording the state and its flux crossing each suspected zone where a source could occur, we establish an identifiability theorem that yields uniqueness of the unknown elements defining all occurring sources. We develop a non-iterative detection–identification method that goes throughout the monitored domain to detect in each suspected zone whether there exists or not an occurring source. Once a source is detected, the developed method determines lower and upper bounds of the total amount discharged by the occurring source and localizes the geometric centre of its unknown spatial support. Then, given two desired reference geometries for example, squares/discs centred at the already localized geometric centre, the method determines the biggest domain defined by a first reference geometry included in the sought source spatial support as well as the smallest domain defined by a second reference geometry containing this unknown support. In addition, the developed method gives an approximation of the surface area of this latest and identifies its time-dependent intensity function. Some numerical experiments on a variant of the surface water Biological Oxygen Demand pollution model are presented.