Non‐linear model reduction for uncertainty quantification in large‐scale inverse problems

We present a model reduction approach to the solution of large‐scale statistical inverse problems in a Bayesian inference setting. A key to the model reduction is an efficient representation of the non‐linear terms in the reduced model. To achieve this, we present a formulation that employs masked projection of the discrete equations; that is, we compute an approximation of the non‐linear term using a select subset of interpolation points. Further, through this formulation we show similarities among the existing techniques of gappy proper orthogonal decomposition, missing point estimation, and empirical interpolation via coefficient‐function approximation. The resulting model reduction methodology is applied to a highly non‐linear combustion problem governed by an advection–diffusion‐reaction partial differential equation (PDE). Our reduced model is used as a surrogate for a finite element discretization of the non‐linear PDE within the Markov chain Monte Carlo sampling employed by the Bayesian inference approach. In two spatial dimensions, we show that this approach yields accurate results while reducing the computational cost by several orders of magnitude. For the full three‐dimensional problem, a forward solve using a reduced model that has high fidelity over the input parameter space is more than two million times faster than the full‐order finite element model, making tractable the solution of the statistical inverse problem that would otherwise require many years of CPU time. Copyright © 2009 John Wiley & Sons, Ltd.     

Three‐dimensional magnetostatic problem

The paper is devoted to an approximation of the solution of Maxwell's equations in three‐dimensional space. We present two methods which couple a finite element method inside the magnetic materials with a boundary integral method which uses Poincaré–Steklov's operator to describe the exterior domain. A computer code has been implemented for each method and a number of numerical experiments have been performed to validate each proposed methodology. Namely, we present numerical results concerning a non‐linear magnetostatic problem in ℝ³. Copyright © 1999 John Wiley & Sons, Ltd.     

Sensitivity analysis and design optimization of three‐dimensional non‐linear aeroelastic systems by the adjoint method

We consider the problem of optimizing a non‐linear aeroelastic system in steady‐state conditions, where the structure is represented by a detailed finite element model, and the aerodynamic loads are predicted by the discretization of the non‐linear Euler equations. We present a solution method for this problem that is based on the three‐field formulation of fluid–structure interaction problems, and the adjoint approach for coupled sensitivity analysis. We discuss the computational complexity of the proposed solution method, describe its implementation on parallel processors, and illustrate its computational efficiency with the aeroelastic optimization of various wings. Copyright © 2002 John Wiley & Sons, Ltd.     

Solving inverse electromagnetic problems using FDTD and gradient‐based minimization

We address time‐domain inverse electromagnetic scattering for determining unknown characteristics of an object from observations of the scattered field. Applications include non‐destructive characterization of media and optimization of material properties, for example, the design of radar absorbing materials. Another application is model reduction where a detailed model of a complex geometry is reduced to a simplified model. The inverse problem is formulated as an optimal control problem where the cost function to be minimized is the difference between the estimated and observed fields, and the control parameters are the unknown object characteristics. The problem is solved in a deterministic gradient‐based optimization algorithm using a parallel 2D FDTD scheme. Highly accurate analytical gradients are computed from the adjoint formulation. The inverse method is applied to the characterization of layered dispersive media and the determination of parameters in subcell models for thin sheets and narrow slots. Copyright © 2006 John Wiley & Sons, Ltd.     

A semi‐analytical locally transversal linearization method for non‐linear dynamical systems

An numeric‐analytical, implicit and local linearization methodology, called the locally transversal linearization (LTL), is developed in the present paper for analyses and simulations of non‐linear oscillators. The LTL principle is based on deriving the locally linearized equations in such a way that the tangent space of the linearized equations transversally intersects that of the given non‐linear dynamical system at that particular point in the state space where the solution vector is sought. For purposes of numerical implementation, two different numerical schemes, namely LTL‐1 and LTL‐2 schemes, based on the LTL methodology are presented. Both LTL‐1 and LTL‐2 procedures finally reduce the given set of non‐linear ordinary differential equations (ODEs) to a set of transcendental algebraic equations valid over a short interval of time or over a short segment of the evolving trajectories as projected on the phase space. While in the LTL‐1 scheme the desired solution vector at a forward time point enters the linearized differential equations as an unknown parameter, in the LTL‐2 scheme a set of unknown residues enters the linearized system as parameters. A limited set of examples involving a few well‐known single‐degree‐of‐freedom (SDOF) non‐linear oscillators indicate that the LTL methodology is capable of accurately predicting many complicated non‐linear response patterns, including limit cycles, quasi‐periodic orbits and even strange attractors. Copyright © 2001 John Wiley & Sons, Ltd.     

Initialization of high‐order accuracy immersed interface CFD solvers using complex CAD geometry

This study concerns the development of a numerical methodology for initializing immersed interface‐based CFD solvers for using complex computer‐aided design (CAD) geometry. CFD solvers with higher‐order discretization stencils require larger stencil widths, which become problematic in regions of space where insufficient mesh resolution is available. This problem becomes especially challenging when convoluted triangulated surface meshes generated from complex solid models are used to initialize the cut‐cells. A pragmatic balance between desired local geometry resolution and numerical accuracy is often required to find a practical solution. Here, a robust iterative fill algorithm is presented that balances geometry resolution with numerical accuracy (via stencil size). Several examples are presented to illustrate the use of this initialization procedure that employs both the original CAD generated triangulated surface mesh, along with a level set representation of the surface to initialize cut‐cells and boundary proximity measures for creation of CFD stencils. Convergence error analysis of surface area and enclosed volumes is first presented to show the effects of fill on the geometry as a function of desired stencil size and grid resolution. The algorithm is then applied to geometrically complex problems using large eddy simulation. Two problems are considered. The first is flow around the Eiffel Tower. The second is a combustion swirler in the context of a design problem. Copyright © 2016 John Wiley & Sons, Ltd.     

An inverse heat conduction problem with heat flux measurements

Using heat flux measurements as additional information to solve inverse heat conduction problems was and is still rarely employed. Lot of disadvantages linked to heat flux measurement specificities (local disturbance, intrusive measurement, lack of knowledge and proficiency, etc.) make people prefer temperature measurements which are well documented and very widespread. Solving inverse heat conduction problems with heat flux measurements is quite different than the one which uses temperatures and need to be investigated deeply. In this work, this problem is approached through the solution of a bioengineering problem consisting in the development of a non‐invasive blood perfusion probe. The effort here is focused on the development of a methodology for the estimation of time‐dependent blood perfusion from heat flux measurements. The physical probe incorporates a thin heat flux sensor, which is placed in contact with the tissue region where the perfusion is to be measured. The sensor records the heat flux due to an imposed thermal event, which is achieved by air flow. A one‐dimensional mathematical model is used to simulate the thermal event occurring at the contact region holding between the probe and the tissue. A combined parameter and function estimation procedure is developed to estimate simultaneously time‐dependent blood perfusion and thermal contact conductance between the probe and the tissue. The robustness of the method was demonstrated through several test cases using simulated data. The presented examples include various functional changes in the time evolution of blood perfusion. Results from this study have shown the feasibility of solving inverse problems with heat flux measurements and the two unknowns are estimated with no a priori information about their functional forms. Copyright © 2006 John Wiley & Sons, Ltd.     

A dual algorithm for the topology optimization of non‐linear elastic structures

Dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of Lagrange multipliers associated with the constraints. To date, dual algorithms have been applied only for linear structures. Here we extend this methodology to the case of non‐linear structures. The perimeter constraint is used to make the topology problem well‐posed. We show that the proposed algorithm yields a value of perimeter that is close to that specified by the user. We also address the issue of manufacturability of these designs, by proposing a variant of the standard dual algorithm, which generates designs that are two‐dimensional although the loading and the geometry are three‐dimensional. Copyright © 2008 John Wiley & Sons, Ltd.     

A new methodology to solve non‐linear equation systems using genetic algorithms. Application to combined cyclegas turbine simulation

This paper shows a methodology to sort out the equations of a non‐linear system in order to solve it by the fixed‐point method. The arrangement of the equations is established by a genetic algorithm that deals with a population of possible resolution processes of the system. The method is specially useful in the following situations: first, when the system is very non‐linear and has many variables (where the Newton–Raphson method does not work properly); second, when the number of equations and variables may be altered because the equation system may change in each simulation and, therefore, more than one only solution process is needed if the fixed‐point process is employed. As an example, the methodology has been applied to solve the equation system that models the behaviour of a heat recovery steam generator of a combined cycle power plant at full load and part load conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

A framework for fracture modelling based on the material forces concept with XFEM kinematics

A theoretical and computational framework which covers both linear and non‐linear fracture behaviour is presented. As a basis for the formulation, we use the material forces concept due to the close relation between on one hand the Eshelby energy–momentum tensor and on the other hand material defects like cracks and material inhomogeneities. By separating the discontinuous displacement from the continuous counterpart in line with the eXtended finite element method (XFEM), we are able to formulate the weak equilibrium in two coupled problems representing the total deformation. However, in contrast to standard XFEM, where the direct motion discontinuity is used to model the crack, we rather formulate an inverse motion discontinuity to model crack development. The resulting formulation thus couples the continuous direct motion to the inverse discontinuous motion, which may be used to simulate linear as well as non‐linear fracture in one and the same formulation. In fact, the linear fracture formulation can be retrieved from the non‐linear cohesive zone formulation simply by confining the cohesive zone to the crack tip. These features are clarified in the two numerical examples which conclude the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

Analysis of SMA composite laminates using a multiscale modelling technique

The present work deals with the analysis of smart laminates, obtained as stacking sequence of fibre‐reinforced composite laminae and composite shape memory alloy (SMA) layers. The behaviour of composite SMA (CSMA) laminate is studied developing a full micro–macro approach. In fact, a non‐linear 4‐node mixed interpolation of tensorial components (MITC4) laminate finite element, based on the first‐order shear deformation theory, is developed. The SMA layer constitutive relationship is determined solving a non‐linear homogenization problem at each non‐linear iteration of each time step for each integration Gauss point. Some numerical applications are developed in order to investigate the influence of the CSMA on the buckling behaviour of plates and on the transversal displacement control of plates subjected to different loading conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

A meshless grain element for micromechanical analysis with crystal plasticity

This study concerns the development of a 2‐D meshless grain element for elasto‐plastic deformation and intergranular damage initiation and propagation in polycrystalline fcc metals under static loading. The crystallographic material behaviour of the grains is represented by a rate‐independent single‐crystal plasticity model while including material orthotropy. The two slip planes are arbitrarily located with respect to the crystallographic axis of the grain. A non‐linear constitutive model known as the cohesive zone model is employed to represent the inelastic interaction between the grain boundaries, thus permitting grain boundary opening and sliding. The cohesive model describes the deformation characteristics of the grain boundaries through a non‐linear relation between the effective grain boundary tractions and displacements. Because of the presence of non‐linear material behaviour both inside the grain and along the cohesive grain boundaries, the method utilizes the principle of virtual work in conjunction with the meshless formulation in the derivation of the system of non‐linear incremental equilibrium equations. The solution is obtained via an incremental procedure based on the Taylor series expansion about the current equilibrium configuration. The fidelity of the present approach is verified by considering simple polycrystalline metals of only a few grains. Copyright © 2006 John Wiley & Sons, Ltd.     

A non‐physical enthalpy method for the numerical solution of isothermal solidification

In this paper, a new formulation for the solution of the discontinuous isothermal solidification problem is presented. The formulation has similarities with the now classical capacitance and source methods traditionally used in commercial software. However, the new approach focuses on the solution of the governing enthalpy‐transport equation rather than the governing parabolic partial differential heat equation. The advantage is that discontinuous physics can be accounted for without approximation and the arbitrariness common to classic approaches is avoided. Also introduced is the concept of non‐physical enthalpy, which unlike physical enthalpy has numerical values that are not moving‐frame invariant. Understanding the behaviour of the non‐physical enthalpy is central to the successful treatment of discontinuities. A particular drawback is that non‐physical enthalpy is non‐intuitive and new mathematical constructs are required to describe its behaviour. This involves the introduction of transport equations, which provide the new concept of relative moving‐frame invariance for the non‐physical enthalpy. The principal advantage, however, is that a unified methodology is established for the treatment of discontinuities. This is shown to establish real rigour and in many respects the formulation highlights the erroneous choices made with established classical approaches and casts in a totally new light a somewhat traditional problem. The new methodology is applied to a range of simple problems not only to provide an in‐depth treatment and for ease of understanding but also to best describe the behaviour of the non‐intuitive non‐physical enthalpy. Demonstrated in the paper is the methods' remarkable accuracy and stability. Copyright © 2010 John Wiley & Sons, Ltd.     

Reduced‐order modeling of parameterized PDEs using time–space‐parameter principal component analysis

This paper presents a methodology for constructing low‐order surrogate models of finite element/finite volume discrete solutions of parameterized steady‐state partial differential equations. The construction of proper orthogonal decomposition modes in both physical space and parameter space allows us to represent high‐dimensional discrete solutions using only a few coefficients. An incremental greedy approach is developed for efficiently tackling problems with high‐dimensional parameter spaces. For numerical experiments and validation, several non‐linear steady‐state convection–diffusion–reaction problems are considered: first in one spatial dimension with two parameters, and then in two spatial dimensions with two and five parameters. In the two‐dimensional spatial case with two parameters, it is shown that a 7 × 7 coefficient matrix is sufficient to accurately reproduce the expected solution, while in the five parameters problem, a 13 × 6 coefficient matrix is shown to reproduce the solution with sufficient accuracy. The proposed methodology is expected to find applications to parameter variation studies, uncertainty analysis, inverse problems and optimal design. Copyright © 2009 John Wiley & Sons, Ltd.     

Efficient optimization processes using kriging approximation models in electrical impedance tomography

A reduced model technique based on kriging approximations is developed in order to increase the rate of convergence of an evolution strategy (ES) when solving a non‐destructive evaluation problem. The inverse problem investigated consists of identifying the geometry of discontinuities in a conductive material from Cauchy data measurements taken on the boundary. In this study, we use kriging approximation models in order to increase the rate of convergence of the optimization algorithm and to efficiently detect, from a computational time point of view, a subsurface cavity, such as a circle. The algorithm developed by combining evolution strategies and kriging approximations is found to be a robust, fast and efficient method for detecting the size and location of subsurface cavities. Copyright © 2006 John Wiley & Sons, Ltd.     

Design rules for autofrettage of an aluminium valve body

The following paper is intended to improve the fatigue behaviour of a complex aluminium valve geometry under high internal cyclic pressure loading. The autofrettage process helps to increase the fatigue durability and a simple, but efficient design method for this process is deployed. Based on non‐linear material's behaviour, finite element simulations of the crack‐free geometry help to determine the minimum and maximum autofrettage pressure to be used, without iterative crack simulations, which would require higher computational effort. Material tests under inverse plastifications were performed in order to determine the correct material model. The derived design method was validated with simplified specimens subjected to different autofrettage pressure levels and subsequent cyclic fatigue tests.     

Self‐regularized pseudo time‐marching schemes for structural system identification with static measurements

We explore the application of pseudo time marching schemes, involving either deterministic integration or stochastic filtering, to solve the inverse problem of parameter identification of large dimensional structural systems from partial and noisy measurements of strictly static response. Solutions of such non‐linear inverse problems could provide useful local stiffness variations and do not have to confront modeling uncertainties in damping, an important, yet inadequately understood, aspect in dynamic system identification problems. The usual method of least‐square solution is through a regularized Gauss–Newton method (GNM) whose results are known to be sensitively dependent on the regularization parameter and data noise intensity. Finite time, recursive integration of the pseudo‐dynamical GNM (PD‐GNM) update equation addresses the major numerical difficulty associated with the near‐zero singular values of the linearized operator and gives results that are not sensitive to the time step of integration. Therefore, we also propose a pseudo‐dynamic stochastic filtering approach for the same problem using a parsimonious representation of states and specifically solve the linearized filtering equations through a pseudo‐dynamic ensemble Kalman filter (PD‐EnKF). For multiple sets of measurements involving various load cases, we expedite the speed of the PD‐EnKF by proposing an inner iteration within every time step. Results using the pseudo‐dynamic strategy obtained through PD‐EnKF and recursive integration are compared with those from the conventional GNM, which prove that the PD‐EnKF is the best performer showing little sensitivity to process noise covariance and yielding reconstructions with less artifacts even when the ensemble size is small. Copyright © 2009 John Wiley & Sons, Ltd.     

Numerical methods for the topology optimization of structures that exhibit snap‐through

The inclusion of non‐linear elastic analyses into the topology optimization problem is necessary to capture the finite deformation response, e.g. the geometric non‐linear response of compliant mechanisms. In previous work, the non‐linear response is computed by standard non‐linear elastic finite element analysis. Here, we incorporate a load–displacement constraint method to traverse non‐linear equilibrium paths with limit points to design structures that exhibit snap‐through behaviour. To accomplish this, we modify the basic arc length algorithm and embed this analysis into the topology optimization problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Experimental evaluation of the elastic limit of carbon‐fibre reinforced epoxy composites under a large range of strain rate and temperature conditions

The mechanical behaviour of organic matrix composite materials such as T700GC/M21 carbon fibre reinforced polymer (CFRP) is generally considered by the industry as being orthotropic elastic for the sizing of aeronautical structures under normal isothermal “static” flight loads. During the aircraft lifetime, it may be exposed to severe loading conditions at various temperatures. However, the mechanical behaviour of CFRP is known to exhibit a linear behaviour or a non‐linear behaviour according to the types of loads that are considered creep or extreme conditions. The observed non‐linearity can be commonly attributed to several physical phenomena such as non‐linear viscosity, plasticity, or damage. In the literature, different models can be found that are based on three components: a first elastic reversible behaviour, a second non‐linear behaviour, and a failure criterion. An important issue is to understand and characterize the transition between the elastic reversible behaviour and the non‐linear behaviour. To answer this question, the present paper describes an experimental methodology that permits to evaluate this transition thanks to raw experimental data, and its application to a range of constant but different strain rate and temperature tests performed on the T700GC/M21 CFRP material.     

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.