A comparison between inverse form finding and shape optimization methods for anisotropic hyperelasticity in logarithmic strain space

In this paper an inverse mechanical formulation and a Limited-Broyden-Flechter-Goldfarb-Shanno method for shape optimization are compared. Both methods deal with the determination of the undeformed shape of an hyperelastic part knowing its deformed configuration and the applied loads. We consider anisotropic hyperelastic materials that are formulated in the logarithmic strain space. Beside the theoretical aspects, we present a numerical example. We established that no difference could be found between the node coordinates on the undeformed sheets computed with both methods. However the convergence to the solution is faster for the inverse mechanical formulation compared to the L-BFGS algorithm. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A comparison between a recursive method based on an inverse mechanical formulation and shape optimization for solving inverse form finding problems in isotropic elastoplasticity

In this paper we compare two methods, a recursive method based on an inverse mechanical formulation and a method based on a recursive shape optimization formulation, in order to solve inverse form finding problems in isotropic elastoplasticity. Both methods are succinctly presented and a numerical example is given. It was found that no difference could be found between the node coordinates on the undeformed configurations computed with both methods. However the convergence to the solution is faster with the recursive method based on an inverse mechanical formulation than with the method based on shape optimization. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimality criteria coupled adaptive mesh method for optimal shape design of Stokes flow

An adaptive mesh method combined with the optimality criteria algorithm is applied to optimal shape design problems of fluid dynamics. The shape sensitivity analysis of the cost functional is derived. The optimization problem is solved by a simple but robust optimality criteria algorithm, and an automatic local adaptive mesh refinement method is proposed. The mesh adaptation, with an indicator based on the material distribution information, is itself shown as a shape or topology optimization problem. Taking advantages of this algorithm, the optimal shape design problem concerning fluid flow can be solved with higher resolution of the interface and a minimum of additional expense. Details on the optimization procedure are provided. Numerical results for two benchmark topology optimization problems are provided and compared with those obtained by other methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimum blank design by using the FEM based on inverse motion in orthotropic elasto-plasticity

Inverse form finding aims to determine the optimum blank design of a workpiece whereby the desired end configuration that is obtaining after a forming process, the boundary conditions and the applied loads are known. Inputting the optimal initial configuration a subsequent computation with the Finite Element Method (FEM) then has to result in exactly the nodal coordinates of the desired deformed workpiece. Germain et al. [1] recently presented a new form finding strategy for isotropic elasto-plasticity, see also Landkammer et al. [2] for orthotropic plasticity. The corresponding algorithm uses the inverse mechanical formulation (also denoted as inverse finite element method) in addition to the common direct formulation in a recursive way. Switching between the direct and the inverse mechanical formulation, while fixing the internal plastic variables in the inverse step, uniquely detects the undeformed configuration iteratively. This contribution demonstrates within an example that the developed recursive algorithm even works with combinations of orthotropic elastic and orthotropic plastic material parameters without affecting the nearly linear convergence rate of the form finding algorithm. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary element method for dynamic inelastic analysis

The paper shows formulation and application of the boundary element method (BEM) for dynamic analysis of elastoplastic materials. The initial stress approach is used in the elastoplastic analysis. The mass matrices are computed by the dual reciprocity method (DRBEM). Displacements and stresses are computed by the iterative procedure in each time step. The time dependent problem is solved by the direct integration Houbolt method. The methods presented in the paper are applied to compute displacements and stresses in a crank loaded by dynamic forces. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical instabilities in structural optimization – analogy between topology & shape design problems

The formulation of structural optimization problems on the basis of the finite–element–method often leads to numerical instabilities resulting in non–optimal designs, which turn out to be difficult to realize from the engineering point of view. In the case of topology optimization problems the formation of designs characterized by oscillating density distributions such as the well–known “checkerboard–patterns” can be observed, whereas the solution of shape optimization problems often results in unfavourable designs with non–smooth boundary shapes caused by high–frequency oscillations of the boundary shape functions. Furthermore a strong dependence of the obtained designs on the finite–element–mesh can be observed in both cases. In this context we have already shown, that the topology design problem can be regularized by penalizing spatial oscillations of the density function by means of a penalty–approach based on the density gradient. In the present paper we apply the idea of problem regularization by penalizing oscillations of the design variable to overcome the numerical difficulties related to the shape design problem, where an analogous approach restricting the boundary surface can be introduced. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Experimental and numerical study of the relaxation behavior of facial soft tissue

Challenges in computational simulation of the mechanical behavior of soft tissues and organs for clinical applications are related to the reliability of the models with respect to the anatomy, the mechanical interactions between different tissues, and the non linear (time dependent) force deformation characteristics of soft biological materials. In this paper a 3D finite element model of the face and neck, which has applications in surgical devices optimization and surgery planning, is presented. Bones, muscles, skin, fat, and superficial muscoloaponeurotic system (SMAS) were reconstructed from magnetic resonance images and their shape, constraints and interactions have been modeled according to anatomical, plastic and reconstructive surgery literature. Non linear time dependent constitutive equations are implemented in the numerical model, based on the Rubin-Bodner model. For the present calculations a simplified hyperelastic formulation has been used. The corresponding model parameters were selected according to previous work with mechanical measurements ex vivo on facial soft tissue. For determination of model parameters, in particular the ones corresponding to the time dependent behavior, an instrument for measuring the relaxation behavior of the face tissue in vivo was developed. The experimental set-up is described and results are presented for tests performed on different locations of the face (jaw, mid-face, parotid regions) and neck. The measured “long term” reaction force of the facial soft tissue is compared to numerical results. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape functional optimization with restrictions boosted with machine learning techniques

Shape optimization is a widely used technique in the design phase of a product. Current ongoing improvement policies require a product to fulfill a series of conditions from the perspective of mechanical resistance, fatigue, natural frequency, impact resistance, etc. All these conditions are translated into equality or inequality restrictions which must be satisfied during the optimization process that is necessary in order to determine the optimal shape. This article describes a new method for shape optimization that considers any regular shape as a possible shape, thereby improving on traditional methods limited to straight profiles or profiles established a priori. Our focus is based on using functional techniques and this approach is, based on representing the shape of the object by means of functions belonging to a finite-dimension functional space. In order to resolve this problem, the article proposes an optimization method that uses machine learning techniques for functional data in order to represent the perimeter of the set of feasible functions and to speed up the process of evaluating the restrictions in each iteration of the algorithm. The results demonstrate that the functional approach produces better results in the shape optimization process and that speeding up the algorithm using machine learning techniques ensures that this approach does not negatively affect design process response times.     

Recent advances in the numerical modeling of constitutive relations

In this paper we present an overview of the recent developments in the area of numerical and finite element modeling of nonlinear constitutive relations. The paper discusses elastic, hyperelastic, elastoplastic and anisotropic plastic material models. In the hyperelastic model an emphasis is given to the method by which the incompressibility constraint is applied. A systematic and general procedure for the numerical treatment of hyperelastic model is presented. In the elastoplastic model both infinitesimal and large strain cases are discussed. Various concerns and implications in extending infinitesimal theories into large strain case are pointed out. In the anisotropic elastoplastic case, emphasis is given to the practicality of proposed theories and its feasible and economical use in the finite element environment.     

Shape optimization of a body in the Oseen flow

This article is concerned with a numerical simulation of shape optimization of the Oseen flow around a solid body. The shape gradient for shape optimization problem in a viscous incompressible flow is computed by the velocity method. The flow is governed by the Oseen equations with mixed boundary conditions containing the pressure. The structure of continuous shape gradient of the cost functional is derived by using the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. A gradient type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose and the proposed algorithm is feasible. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

MULTI-SCALE METHODS FOR INVERSE MODELING IN 1-D MOS CAPACITOR

In this paper,we investigate multi-scale methods for the inverse modeling in 1-D Metal-Oxide-Silicon(MOS) capactior,First,the mathematical model of the device is given and the numerical simulation for the forward problem of the model is implemented using finite element method with adaptive moving mesh. Then numerical analysis of these parameters in the model for the inverse problems is presented .Some matrix analysis tools are applied to explore the parameters‘ sensitivities,And thired,the parameters are extracted using Levenberg-Marquardt optimization method.The essential difficulty arises from the effect of multi-scale physical differeence of the parameters.We explore the relationship between the parameters‘ sensitivitites and the sequencs for optimization,which can seriously affect the final inverse modeling results.An optimal sequence can efficiently overcome the multip-scale problem of these parameters,Numerical experiments show the efficiency of the proposed methods.     

Preconditioning the Pressure Tracking in Fluid Dynamics by Shape Hessian Information

Potential flow pressure matching is a classical inverse design aerodynamic problem. The resulting loss of regularity during the optimization poses challenges for shape optimization with normal perturbation of the surface mesh nodes. Smoothness is not enforced by the parameterization but by a proper choice of the scalar product based on the shape Hessian, which is derived in local coordinates for starshaped domains. Significant parts of the Hessian are identified and combined with an aerodynamic panel solver. The resulting shape Hessian preconditioner is shown to lead to superior convergence properties of the resulting optimization method. Additionally, preconditioning gives the potential for level independent convergence.     

基于Cosserat连续体理论的粉末高温合金弹塑性损伤分析

回顾了航空发动机涡轮盘粉末高温合金材料的发展及研究方法,基于Cosserat连续体理论,建立了粉末高温合金材料的弹塑性损伤模型,可通过特征长度考虑材料的微结构特征,并在模拟软化问题时消除局部化带的网格依赖性.在软化问题中,经典弹塑性理论在计算时需要较多迭代,有时甚至不能收敛.该文基于参变量变分原理,把原非线性问题转化为互补问题来求解,可大大提高求解效率和收敛性.最后通过数值算例验证了本文提出方法的有效性.     

Shape and topology optimization of contact problems by level set methods

This paper deals with the numerical solution of a topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is considered. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. In the paper shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization. Level set based numerical algorithm for the solution of the shape optimization problem is proposed. Level set method is used to describe the position of the boundary of the body and its evolution on a fixed mesh. This evolution is governed by Hamilton – Jacobi equation. The speed vector field driving the propagation of the boundary of the body is given by the shape derivative of a cost functional with respect to the free boundary. Numerical examples are provided. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

We compute a local linearization for the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton‐type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.     

A global damping factor for a non-invasive form finding algorithm

Inverse form finding based on the finite element method (FEM) aims in determining the optimal material (undeformed) configuration when knowing the target spatial (deformed) configuration in a discretized setting. The strategy is to iteratively update the material coordinates and recompute the spatial configuration by a FEM simulation until the computed spatial nodal positions are close enough to a priori given spatial nodal positions. A form finding algorithm is utilized, which is purely based on geometrical considerations and can be coupled with arbitrary external FEM software via subroutines in a non-invasive fashion. At large deformations degenerated elements can occur when updating the material coordinates. Evaluating the mesh quality of the updated material configuration and adjusting a global damping factor before recomputing the next spatial configuration helps to avoid mesh distortions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Textile porosity determination based on a nonlinear heat and moisture transfer model

The inverse problems of textile materials design on heat and moisture transfer properties are important and indispensable in applications in the body-clothing-environment system. We present an inverse problem of textile porosity determination (IPTPD) based on a nonlinear heat and moisture transfer model. Adopting the idea of the least-squares, the mathematical formulation of IPTPD is deduced to a regularized optimization problem with collocation method applied. The continuity of the regularized minimization problem is proved. By means of genetic algorithm (GA), the approximate solution of the IPTPD is numerically obtained. To reduce the computational cost, an improved algorithm based on BP neural network with GA is proposed in the numerical simulation. Compared with the direct GA searching, the computational cost is greatly reduced, which presents a similar result.     

Vortex design problem

In this investigation we propose a computational approach for the solution of optimal control problems for vortex systems with compactly supported vorticity. The problem is formulated as a PDE-constrained optimization in which the solutions are found using a gradient-based descent method. Recognizing such Euler flows as free-boundary problems, the proposed approach relies on shape differentiation combined with adjoint analysis to determine cost functional gradients. In explicit tracking of interfaces (vortex boundaries) this method offers an alternative to grid-based techniques, such as the level-set methods, and represents a natural optimization formulation for vortex problems computed using the contour dynamics technique. We develop and validate this approach using the design of 2D equilibrium Euler flows with finite-area vortices as a model problem. It is also discussed how the proposed methodology can be applied to Euler flows featuring other vorticity distributions, such as vortex sheets, and to time-dependent phenomena.     

Plateau regularization method for structural shape optimization and geometric mesh control

Within this contribution the algorithmic treatment of the inverse problem of finding mechanically motivated membrane shapes is discussed and the key point, that the corresponding numerical methods have a broader spectrum of applications which enables their adoption to similar problems from other disciplines, is highlighted. The presented adaptive scheme is the Updated Reference Strategy (URS) enhanced by a newly derived “element distortion control”. The key feature is the ability of the proposed stabilized scheme to overcome the singular problem of finding equilibrium shapes of prestressed membranes which is due to the non–uniqueness of nodal positions in the finite element mesh and the purposeful adjustment of the underlying stress state based on a local geometrical criterion in case of incompatible stress states. Therefore, the derived methodology is –beside the mere computation of equilibrium configurations– able to distribute (probably very local) deformations of the mesh in a smooth way to the whole domain by at the same time conserving specific mesh characteristics which guarantees regular meshes with high quality concerning the element shape. This results in robust computations even for complex and problematic geometrical situations. Due to the effective mesh control of this approach a transfer of the developed methodology to other fields of application like e.g. mesh smoothing, large displacement mesh moving problems and stabilized CAD–free shape optimization of shells is promising and therefore accomplished. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shape Methods for the Transmission Problem with a Single Measurement

In the current work, we consider the inverse conductivity problem of recovering inclusion with one measurement. First, we use conformal mapping techniques for determining the location of the anomaly and estimating its size. We then get a good initial guess for quasi-Newton type method. The inverse problem is treated from the shape optimization point of view. We give a rigorous proof for the existence of the derivative of the state function and of shape functionals. We consider both least squares fitting and Kohn and Vogelius functionals. For the numerical implementation, we use a parameterization of shapes coupled with a boundary element method. Several numerical examples indicate the superiority of the Kohn and Vogelius functional over least squares fitting.