Inverse Geometric Problem Research Articles

Optimal design of multi-layer thermal protection of variable thickness

PurposeThe presented paper aims to consider algorithm for optimal design of multilayer thermal insulation.Design/methodology/approachDeveloped algorithm is based on a sequential quadratic programming method.Findings2D mathematical model of heat transfer in thermal protection was considered in frame of thermal design of spacecraft. The sensitivity functions were used to estimate the Jacobean of the object functions.Research limitations/implicationsDesign of distributed parameter systems and shape optimization may be thought of as geometrical inverse problems, in which the positions of free boundaries are determined along with the spatial variables. In such problems, the missing data (i.e. the position of boundaries) are compensated for by the presence of the so-called inverse problem additional conditions. In the case under consideration, such conditions are constrains on the temperature values at the discrete points of the system.Practical implicationsResults are presented how to apply the algorithm suggested for solving a practical problem – thickness sampling for a thermal protection system of advanced solar probe.Originality/valueThe procedure proposed in the paper to solve a design problem is based on the method of quadratic approximation of the initial problem statement as a Lagrange formulation. This has allowed to construct a rather universal algorithm applicable without modification for solving a wide range of thermal design problems.

An energy gap functional: Cavity identification in linear elasticity

Abstract The aim of this work is an analysis of some geometrical inverse problems related to the identification of cavities in linear elasticity framework. We rephrase the inverse problem into a shape optimization one using an energetic least-squares functional. The shape derivative of this cost functional is combined with the level set method in a steepest descent algorithm to solve the shape optimization problem. The efficiency of this approach is illustrated by several numerical results.

Automated hybrid singularity superposition and anchored grid pattern BEM algorithm for the solution of inverse geometric problems

A method for solving an inverse geometric problem is presented by reconstructing the unknown subsurface cavity geometry with the boundary element method (BEM) and a genetic algorithm in combination with the Nelder-Mead non-linear simplex optimization method. The heat conduction problem is solved by the BEM which calculates the difference between the measured temperature at the exposed surface and the computed temperature under the current update of the unknown subsurface flaws and cavities. In a first step, clusters of singularities are utilized to solve the inverse problem and to identify the location of the centroid(s) of the subsurface cavity(ies)/flaw(s). In a second step, the reconstruction of the estimated cavity(ies)/flaw(s) geometry(ies) is accomplished by utilizing an anchored grid pattern upon which cubic spline knots are restricted to move in the search for the unknown geometry. The solution is achieved using a genetic algorithm accelerated with the Nelder-Mead non-linear simplex method. The automated algorithm successfully reconstructs single and multiple subsurface cavities within two dimensional mediums. The cavity detection was enhanced by applying multiple boundary condition sets (MBCS) to the same geometry. This extra information supplied on the boundary made the subsurface cavity easily detectable despite its low heat signature effect on the boundaries.

A Bayesian level set method for geometric inverse problems

We introduce a level set based approach to Bayesian geometric inverse problems. In these problems the interface between different domains is the key unknown, and is realized as the level set of a function. This function itself becomes the object of the inference. Whilst the level set methodology has been widely used for the solution of geometric inverse problems, the Bayesian formulation that we develop here contains two significant advances: firstly it leads to a well-posed inverse problem in which the posterior distribution is Lipschitz with respect to the observed data, and may be used to not only estimate interface locations, but quantify uncertainty in them; and secondly it leads to computationally expedient algorithms in which the level set itself is updated implicitly via the MCMC methodology applied to the level set function – no explicit velocity field is required for the level set interface. Applications are numerous and include medical imaging, modelling of subsurface formations and the inverse source problem; our theory is illustrated with computational results involving the last two applications.

Factorization method in the geometric inverse problem of static elasticity

The factorization method, which has previously been used to solve inverse scattering problems, is generalized to geometric inverse problems of static elasticity. We prove that finitely many defects (cavities, cracks, and inclusions) in an isotropic linearly elastic body can be determined uniquely if the operator that takes the forces applied to the body outer boundary to the outer boundary displacements due to these forces is known.

Reconstruction of defects in elastic bodies by combination of genetic algorithm and finite element method

Modeling of the non-destructive testing system of defects in solids is performed. Specifically, the inverse geometric problems of the elasticity theory for a flat rectangular area on reconstructing circular cavities and cracks breaking the body surface are considered. Additional information for solving these problems is a setting of the first four natural resonance frequencies. The inverse problem solution is based on the minimization of the residual functional between the measured input source information and the data calculated during the numerical solution of direct problems with the given parameters of defects. As a tool for solving direct problems, the finite element method implemented in FlexPDE program is used. The functional minimization is carried out by using a genetic algorithm (GA) implemented in the developed GAFEMNDT program. The program algorithm and GA settings used in the numerical experiments are described. The experiments results on determining parameters of defects (coordinates of centre, radius, coordinates of surface cracking and its size) are presented. The results demonstrate adequacy of the additional information to overcome the problem ill-posedness, as well as high efficiency of the proposed algorithm both in accuracy of detecting defects parameters, and in their search speed.

Identification of the geometry and elastic properties of rigid inclusions in thin plate

The presence of inclusions in thin-walled structural elements in the course of their operation under mechanical and thermal loads gives rise to the material damage. In publications on this subject, the problem of defect identification is considered in two directions – the location and shape of the defect or the physical properties of rigid inclusions are determined. The paper proposes a method for simultaneous determination of the location of rigid inclusions and their elastic properties (Young's modulus) in a thin plate. The mathematical model of a plate with the inclusion is built in the framework of the linear theory of elasticity. For quantization of unknown functions, the finite element method is used. The geometric inverse problem is formulated in the conditional-correct statement. The parameters of rigid inclusions are determined by minimizing the quality functional. Additional conditions are attached to the functional using the Lagrange multipliers. The results of the identification of one or several inclusions of different sizes by the proposed method are presented. In solving the problem for a plate with one rigid inclusion, the error of restoration of the inclusion location does not exceed 3 %, the error of determination of the Young's modulus of this inclusion – 2 %. The error of determination of the Young's modulus for a group of several rigid inclusions is in the range of 4–7 %. Comparative analysis of numerical experiment results shows that the proposed method allows identifying the location and properties of several rigid inclusions of various sizes. Solution of the problem of identifying the inclusions, determining their size and elastic properties allows more accurate estimation of the strength characteristics of elements and prediction of the service life of structures.

The method of fundamental solutions for three-dimensional inverse geometric elasticity problems

We investigate the numerical reconstruction of smooth star-shaped voids (rigid inclusions and cavities) which are compactly contained in a three-dimensional isotropic linear elastic medium from a single set of Cauchy data (i.e. nondestructive boundary displacement and traction measurements) on the accessible outer boundary. This inverse geometric problem in three-dimensional elasticity is approximated using the method of fundamental solutions (MFS). The parameters describing the boundary of the unknown void, its centre, and the contraction and dilation factors employed for selecting the fictitious surfaces where the MFS sources are to be positioned, are taken as unknowns of the problem. In this way, the original inverse geometric problem is reduced to finding the minimum of a nonlinear least-squares functional that measures the difference between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial coordinates describing the position of the star-shaped void. The interior source points are anchored and move with the void during the iterative reconstruction procedure. The feasibility of this new method is illustrated in several numerical examples.

Cavities Identification from Partially Overdetermined Boundary Data in Linear Elasticity

Our main interest in this work is an analysis of geometrical inverse problem related to the detection of cavities, in elasticity framework from partially overdetermined boundary data in two spatial dimensions. For the reconstruction, we have only access to the displacement field and to the normal component of the normal stress. We propose an identification method based on the Kohn-Vogelius formulation combined with the topological gradient method. An asymptotic expansion for an energy function is derived with respect to the creation of a small hole. A one-shot reconstruction algorithm based on the topological sensitivity analysis is implemented. Some numerical experiments concerning the cavities identification are finally reported, highlighting the ability of the method to identify multiple cavities.

On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

Abstract We consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurements of the Cauchy force on the exterior boundary. We deal with the case where the fluid equations are the nonstationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid-structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier–Stokes system and a fluid-structure system with free boundary. Using complex spherical waves, we obtain some partial information on the distance from a point to the obstacle.

INVERSE VOF MESHLESS METHOD FOR EFFICIENT NONDESTRUCTIVE THERMOGRAPHIC EVALUATION

A novel computational tool based on the Localized Radial-basis Function (RBF) Collocation (LRC) Meshless method coupled with a Volume-of- Fluid (VoF) scheme capable of accurately and efficiently solving transient multi-dimensional heat conduction problems in composite and heterogeneous media is formulated and implemented. While the LRC Meshless method lends its inherent advantages of spectral convergence and ease of automation, the VoF scheme allows to effectively and efficiently simulate the location, size, and shape of cavities, voids, inclusions, defects, or de-attachments in the conducting media without the need to regenerate point distributions, boundaries, or interpolation matrices. To this end, the Inverse Geometric problem of Cavity Detection can be formulated as an optimization problem that minimizes an objective function that computes the deviation of measured temperatures at accessible locations to those generated by the LRC-VoF Meshless method. The LRC-VoF Meshless algorithms will be driven by an optimization code based on the Simplex Linear Programming algorithm which can efficiently search for the optimal set of design parameters (location, size, shape, etc.) within a predefined design space. Initial guesses to the search algorithm will be provided by the classical 1D semi-infinite composite analytical solution which can predict the approximate location of the cavity. The LRC-VoF formulation is tested and validated through a series of controlled numerical experiments. The proposed approach will allow solving the onerous computational inverse geometric problem in a very efficient and robust manner while affording its implementation in modest computational platforms, thereby realizing the disruptive potential of the proposed multi- dimensional high-fidelity non-destructive evaluation (NDE) method.

Identification of rigid inclusions in the thin plate

The problem of identifying the location of the rigid inclusions in thin plate is considered. The finite element method is being used for discretization the unknown functions of the mathematical model. Geometric inverse problem is formulated in the conditional-correct statement, taking into account restrictions on the number of solutions. Parameters identification is performed via minimization of the Lagrangian.

Algorithms for Solution of Direct and Inverse Geometric Problems Involving Determination of Angular Platform Orientation

A six-DOF mechanism of a Steward (or Gough–Steward) type of platform is isolated among parallel-structure mechanisms. Steward [1] and Gough and Whitehall [2] propose a specific application of platforms in special test benches. The general theory of multiple-DOF platforms was developed in the 1980s–1990s [3–9]; however, since then serious investigations have been performed on this subject [10–21]. Analytical methods of solving analytical problems of the statics, kinematics, and dynamics of six-DOF platforms make it possible to establish qualitative properties. Computer programs are currently employed for practical solution of mechanics problems of these platforms. The direct geometry problem for the parallel-structure mechanisms under consideration is formulated in the following manner: the position of the platform (a free solid body with six degrees of freedom in space) is assigned by six geometric parameters, and the platform is hinge-connected to a stationary base with rigid driving limbs equipped with linear drive for varying their lengths (within certain limits). In classical studies involving the mechanics of six-DOF platforms, it is assumed that all degrees of freedom on the stationary base are situated in the same plane, and the hinges on the moving platform are arranged in a single moving plane. In certain structures (for example, [18]), these restrictions are removed, and the hinges on the stationary base and moving platform may be arranged at the whim of the investigator. Problems involving the selection of the position of the spherical hinges of the driving limbs on the stationary base and platform are solved separately on the basis of different requirements (for example, provision for maximum stiffness of the structure on the whole, leaving zones free, etc.); it is then considered that arrangement of the hinges is assigned. Basic Geometric Relationships. It is necessary and sufficient that there be six driving limbs for single-valued assignment of the position of the moving platform on the one hand, and for static determinacy on the other. In the direct geometry (or kinematics) problem, values of the lengths li (distance between centers of spherical hinges) of the six limbs of variChemical and Petroleum Engineering, Vol. 50, Nos. 9–10, January, 2015 (Russian Original Nos. 9–10, Sept.–Oct., 2014)

Some geometric inverse problems for the linear wave equation

In this paper we consider some geometric inverse problems for the linear wave equation. We prove uniqueness results, we present some reconstruction algorithms and we perform numerical experiments in dimensions one and two.

Well-posed Bayesian geometric inverse problems arising in subsurface flow

In this paper, we consider the inverse problem of determining the permeability of the subsurface from hydraulic head measurements, within the framework of a steady Darcy model of groundwater flow. We study geometrically defined prior permeability fields, which admit layered, fault and channel structures, in order to mimic realistic subsurface features; within each layer we adopt either a constant or continuous function representation of the permeability. This prior model leads to a parameter identification problem for a finite number of unknown parameters determining the geometry, together with either a finite number of permeability values (in the constant case) or a finite number of fields (in the continuous function case). We adopt a Bayesian framework showing the existence and well-posedness of the posterior distribution. We also introduce novel Markov chain Monte Carlo (MCMC) methods, which exploit the different character of the geometric and permeability parameters, and build on recent advances in function space MCMC. These algorithms provide rigorous estimates of the permeability, as well as the uncertainty associated with it, and only require forward model evaluations. No adjoint solvers are required and hence the methodology is applicable to black-box forward models. We then use these methods to explore the posterior and to illustrate the methodology with numerical experiments.

Simultaneous numerical determination of a corroded boundary and its admittance

In this paper, an inverse geometric problem for Laplace’s equation arising in boundary corrosion detection is considered. This problem, which consists of determining an unknown corroded portion of the boundary of a bounded domain and its admittance Robin coefficient from two pairs of boundary Cauchy data (boundary temperature and heat flux), is solved numerically using the meshless method of fundamental solutions. A non-linear minimization of the objective function is regularized, and the stability of the numerical results is investigated with respect to noise in the input data and various values of the regularization parameters involved.

Sensitivity analysis for some inverse problems in linear elasticity via minimax differentiability

In this article, we consider the inverse problem of recovering a piecewise constant Lamé parameters by a single boundary measurement. We also consider the geometric inverse problem of locating the interface where the jump of the parameters occurs. These problems turn out to an optimization problems by making use of the Kohn–Vogelius cost function. We rewrite the functional in a min–sup form and we use the differentiability of the min–sup combined with the function space parametrization and the function space embedding to get the optimality condition. These techniques allow us to avoid the differentiability of the states variables with respect to the shape or the Lamé parameters. We apply an iterative algorithm and we give some numerical results.

Line segment cracks identification via the reciprocity gap principle and Fourier transform

The problem of determining a crack by overspecified boundary data is considered. When complete data are avaible on the external boundary.A link that is established between the Reciprocity gap functional and the Fourier transform of the temperature is introduced.If the crack is known (or assumed)to be line, an explicit inversion formulae is obtained and determination of the host line equation and the length of the crack in the two-dimensional (2D) situation. Numerical tests of the identification methods proposed show very good accuracies and significant computational costs. Keywords: Cauchy problem, Inverse geometric problem,the reciprocity gap principle,identification of crack, Fourier Transform,finite element method.

A moving pseudo-boundary MFS for void detection in two-dimensional thermoelasticity

We investigate the numerical reconstruction of smooth star-shaped voids contained in a two-dimensional isotropic linear thermoelastic material from the knowledge of a single non-destructive measurement of the temperature field and both the displacement and traction vectors (over-specified boundary data) on the outer boundary. The primary fields, namely the temperature and the displacement vector, which satisfy the thermo-mechanical equilibrium equations of linear thermoelasticity, are approximated using the method of fundamental solutions (MFS) in conjunction with a two-dimensional particular solution. The fictitious source points are located both outside the (known) exterior boundary of the body and inside the (unknown) void. The inverse geometric problem reduces to finding the minimum of a nonlinear least-squares functional that measures the gap between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial polar coordinates describing the position of the star-shaped void. The interior source points are anchored to and move with the void during the iterative reconstruction procedure. The stability of the numerical method is investigated by inverting measurements contaminated with noise.

An inverse geometric problem: Position and shape identification of inclusions in a conductor domain

Abstract This work presents a methodology for identifying inclusions in a conductor domain. The methodology is based on electrical potential measurements on the external boundary of a conductor body subjected to a prescribed set of electrical current injections. The boundary of each inclusion is approximated by a special kind of spline, whose control points have an extra parameter related to the distance between the control point and the curve. Such special feature allows identification of smooth or sharp inclusions with the same initial guess. The identification is an inverse problem that, in this work, is solved by the Levenberg–Marquardt Method. This iterative method tries to locate the minimum of an objective function, the square of the norm of a residual vector function, given by the differences between electrical potential measurements and the computed ones. The computation of the electrical potential is called forward problem and it is solved by an implementation of the direct formulation of the Boundary Element Method (BEM). The present paper addresses two approaches for computing the derivatives of the residual function with respect to the minimization parameters, required by the Levenberg–Marquardt Method. The first one is based on finite differences and the second one is based on the direct differentiation of the integral equation of the BEM for potential problems. Performance comparisons of these two approaches are presented, based on numerical experiments of identification of inclusions with noisy datasets computationally obtained.