Shape Derivative Research Articles

Topology optimization of compliant mechanism and its support through a level set method

In the present paper, the task of designing a compliant mechanism is accomplished by solving a multi-objective topology optimization problem. The objective function includes a target displacement at the output port and compliances of two load cases. The design variables include two types of boundaries, i.e., the traction free boundary and the Dirichlet boundary. The topology optimization of compliant mechanism and its support is solved by using a level set method. Two level set functions are used to represent a compliant mechanism and the two types of boundaries. Each of the two boundaries is represented by a part of the zero level set of one level set function. Evolution of the two boundaries is modeled by two independent Hamilton–Jacobi equations, thus the shape and topology of the two boundaries change independently. The shape derivatives are derived by using a Lagrangian function and the adjoint method. The finite element analysis is done by modifying a fixed background mesh, and the artificial weak material is not used. Numerical examples in two dimensions are investigated.

Robust shape optimization of continuous structures via the level set method

This work proposes a stochastic shape optimization method for continuous structures using the level-set method. Such a method aims to minimize the expected compliance and its variance as measures of the structural robustness. The behavior of continuous structures is modeled by linear elasticity equations with uncertain loading and material. This uncertainty can be modeled using random variables with different probability distributions as well as random fields. The proper problem formulation is ensured by the proof of the existence colorrev of solution under certain geometrical constraints on the set of admissible shapes. The proposed method addresses the stochastic linear elasticity problem in its weak form obtaining the explicit expressions for the continuous shape derivatives. Some numerical examples are presented to show the effectiveness of the proposed approach.

Estimates of first and second order shape derivatives in nonsmooth multidimensional domains and applications

In this paper we investigate continuity properties of first and second order shape derivatives of functionals depending on second order elliptic PDEs around nonsmooth domains, essentially either Lipschitz or convex, or satisfying a uniform exterior ball condition. We prove rather sharp continuity results for these shape derivatives with respect to Sobolev norms of the boundary-traces of the displacements. With respect to previous results of this kind, the approach is quite different and is valid in any dimension N ≥ 2 . It is based on sharp regularity results for Poisson-type equations in such nonsmooth domains. We also enlarge the class of functionals and PDEs for which these estimates apply. Applications are given to qualitative properties of shape optimization problems under convexity constraints for the variable domains or their complement.

The first eigenvalue of the $$p-$$ p - Laplacian on quantum graphs

We study the first eigenvalue of the $p-$Laplacian (with $1<p<\infty$) on a quantum graph with Dirichlet or Kirchoff boundary conditions on the nodes. We find lower and upper bounds for this eigenvalue when we prescribe the total sum of the lengths of the edges and the number of Dirichlet nodes of the graph. Also we find a formula for the shape derivative of the first eigenvalue (assuming that it is simple) when we perturb the graph by changing the length of an edge. Finally, we study in detail the limit cases $p\to \infty$ and $p\to 1$.

Efficient PDE Constrained Shape Optimization Based on Steklov--Poincaré-Type Metrics

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which rather require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov-Poincar\'e type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor, so far involved in the derivation of shape derivatives.

A Variational Method for Second Order Shape Derivatives

We consider shape functionals obtained as minima on Sobolev spaces of classical integrals having smooth and convex densities, under mixed Dirichlet-Neumann boundary conditions. We propose a new approach for the computation of the second order shape derivative of such functionals, yielding a general existence and representation theorem. In particular, we consider the p-torsional rigidity functional for p grater than or equal to 2.

Assessment of Morphologically Altered RBCs using Image Processing Tools

Morphological study of Red Blood Cell is still reliant on microscopic imaging followed by manual counting, which is associated with personal bias, time and labour. In order to overcome these drawbacks, RBC morphology is quantified using image analysis tool. Because of being objective, reliable and reproducible, comparison of cell number between specimens is considerably more accurate and time saving (2 to 10 times lesser) with automatic counting than with manual counting. In this present work, normal and poikilocytic agent treated samples have been analyzed by microscopy and respective number of shape derivatives was calculated using standard image analysis protocols.

Detection of multiple flaws in piezoelectric structures using XFEM and level sets

Abstract An iterative procedure to solve the inverse problem of detecting multiple voids in piezoelectric structure is proposed. In each iteration the forward problem is solved for various void configurations, and at each iteration, the mechanical and electrical responses of a piezoelectric structure is minimized at known specific points along the boundary to match the measured data. The Extended Finite Element method (XFEM) is employed for determining the responses as it allows the use of a fixed mesh for varying void geometries. The numerical method based on combination of classical shape derivative and of the level-set method for front propagation used in structural optimization is utilized to minimize the cost function. The results obtained show that this method is effectively able to determine the number of voids in a piezoelectric structure and its corresponding locations and shapes.

DAMAGE DETECTION IN LARGE STRUCTURES USING MODES SHAPES AND ITS DERIVATIVES

Synopsis: Many techniques for structural damage detection using modal parameters have been developed over few years because the modal parameters are easily obtained from free, forced and ambient conditions. Many of the techniques have shown that mode shape curvature is efficient to localize damage in continuous system. Focusing on the damage detection/localization of typical Indian Railway bridges, in this paper, a numerical model of an existing steel truss Indian Railway Bridge (discrete system) was developed to check the validity of the mathematical model for continuous shear beam. Numerical model was based on the design drawings of the structure and Eigen value analysis was performed to extract modal parameters. Damage was simulated by modifying the member properties of particular truss members. Results demonstrate that change in fundamental mode shape due damage to a member is an excellent indicator of damage localization because it is discontinuous at damage location. Change in higher derivatives (slope and curvature) of the fundamental mode shape is found to have improved sensitivity in damage localization. These results are similar to those of the shear beams. However, the damage signature is different. Further studies are needed to understand such signatures for meaningful application of the method for existing structures.

Shape optimization of flow field improving hydrodynamic stability

This paper presents a solution of a shape optimization problem of a flow field for delaying transition from a laminar flow to a turbulent flow. Mapping from an initial domain to a new domain is chosen as the design variable. Main problems are defined by the stationary Navier–Stokes problem and an eigenvalue problem assuming a linear disturbance on the solution of the stationary Navier–Stokes problem. The maximum value of the real part of the eigenvalue is used as an objective cost function. The shape derivative of the cost function is defined as the Frechet derivative of the cost function with respect to arbitrary variation of the design variable, which denotes the domain variation, and is evaluated using the Lagrange multiplier method. To obtain a numerical solution, we use an iterative algorithm based on the \(H^{1}\) gradient method using the finite element method. To confirm the validity of the solution, a numerical example for two-dimensional Poiseuille flow with a sudden expansion is presented. Results reveal that a critical Reynolds number increases by the iteration of reshaping.

Shape derivatives for the compressible Navier–Stokes equations in variational form

Shape optimization based on surface gradients and the Hadamard-form is considered for a compressible viscous fluid. Special attention is given to the difference between the “function composition” approach involving local shape derivatives and an alternate methodology based on the weak form of the state equation. The resulting gradient expressions are found to be equal only if the existence of a strong form solution is assumed. Surface shape derivatives based on both formulations are implemented within a Discontinuous Galerkin flow solver of variable order. The gradient expression stemming from the variational approach is found to give superior accuracy when compared to finite differences.

Damage identification in plate-like structure using mode shape derivatives

In this paper, a new damage indicator based on modal data, such as mode shapes and its derivatives, is presented for damage identification in plate-like structures. The proposed indicator is determined using modal analysis information extracted from a finite element code in MATLAB. After obtaining the mode shapes, the slope and curvature of the plate in each mode are calculated based on central finite difference methods. A numerical example with and without noise is considered to evaluate the exact location of different damage scenarios. In order to validate the proposed indicator for structural damage detection, the obtained results have been compared with another study which was based on experimental data. Moreover, in order to better assess the performance of the proposed indicator, a comparison has been made between the proposed indicator and two well-known indicators found in the literature. The results indicate that the proposed damaged indicator is able to detect precisely the location of single and multiple damage cases having different characteristics in plate-like structures.

Shape optimization of a breakwater

In this paper, we optimize the shape of a breakwater which protects a harbour basin from incoming waves. More specifically, our objective is reducing the harbour resonance due to long-range ocean waves. We consider the complex-valued Helmholtz equation as our model state equation and minimize the average wave height in the harbour basin with the shape of the breakwater as optimization variable. The geometry is described by the level set method, i.e. the domain is given as the subzero level set of a function. In contrast to many publications we use the volume expression of the shape derivative, which lends itself naturally to a level set update via a transport equation. The model problem features intrinsic geometric constraints which we treat in the form of forbidden regions. We guarantee feasibility of the iterates by projecting the gradient onto a suitable admissible set.

A shape calculus based method for a transmission problem with a random interface

The present work is devoted to an approximation of the statistical moments of the solution of a class of elliptic transmission problems in R3 with uncertainly located transmission interfaces. In this model, the diffusion coefficient has a jump discontinuity across the random transmission interface which models linear diffusion in two different media separated by an uncertain surface. We apply the shape calculus approach to approximate the solution perturbation by the so-called shape derivative. Correspondingly, statistical moments of the solution are approximated by the moments of the shape derivative. We characterize the shape derivative as a solution of a related homogeneous transmission problem with nonzero jump conditions, which is solved by the boundary integral equation method. A rigorous theoretical framework is developed, and the theoretical findings are supported by and illustrated in two particular examples.

Complex finite element sensitivity method for creep analysis

The complex finite element method (ZFEM) has been extended to perform sensitivity analysis for mechanical and structural systems undergoing creep deformation. ZFEM uses a complex finite element formulation to provide shape, material, and loading derivatives of the system response, providing an insight into the essential factors which control the behavior of the system as a function of time. A complex variable-based quadrilateral user element (UEL) subroutine implementing the power law creep constitutive formulation was incorporated within the Abaqus commercial finite element software. The results of the complex finite element computations were verified by comparing them to the reference solution for the steady-state creep problem of a thick-walled cylinder in the power law creep range. A practical application of the ZFEM implementation to creep deformation analysis is the calculation of the skeletal point of a notched bar test from a single ZFEM run. In contrast, the standard finite element procedure requires multiple runs. The value of the skeletal point is that it identifies the location where the stress state is accurate, regardless of the certainty of the creep material properties.

Shape sensitivities for an inverse problem in magnetic induction tomography based on the eddy current model

In this paper the shape derivative of an objective depending on the solution of an eddy current approximation of Maxwell’s equations is obtained. Using a Lagrangian approach in the spirit of Delfour and Zolésio, the computation of the shape derivative of the solution of the state equation is bypassed. This theoretical result is applied to magnetic impedance tomography, which is an imaging modality aiming at the contactless mapping (identification) of the unknown electrical conductivities inside an object given measurements recorded by receiver coils.

A few shape optimization results for a biharmonic Steklov problem

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue.

Optimal design in elasticity: a systematic adjoint approach to boundary cost functionals

We focus on a particular class of optimal design problems in elasticity where the objective function depends on the stresses along the boundary to be optimized. This is an issue of interest with many engineering applications. This work describes a gradient-type method to solve the finite element numerical approximation of such problems. The descent direction of the discrete cost functional is computed as a discrete projection of the continuous gradient formula, which is derived systematically from shape derivatives and a careful local coordinates calculus. The computational procedure is presented together with an illustrative example, namely the optimization of a cross-sectional tunnel vault immersed in a linearly elastic terrain to obtain uniform compression along the vault.

On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method

The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.

Control of crack propagation by shape-topological optimization

An elastic body weakened by small cracks is considered in the framework of unilateral variational problems in linearized elasticity. The frictionless contact conditions are prescribed on the crack lips in two spatial dimensions, or on the crack faces in three spatial dimensions. The weak solutions of the equilibrium boundary value problem for the elasticity problem are determined by minimization of the energy functional over the cone of admissible displacements. The associated elastic energy functional evaluated for the weak solutions is considered for the purpose of control of crack propagation. The singularities of the elastic displacement field at the crack front are characterized by the shape derivatives of the elastic energy with respect to the crack shape within the Griffith theory. The first order shape derivative of the elastic energy functional with respect to the crack shape, i.e., evaluated for a deformation field supported in an open neighbourhood of one of crack tips, is called the Griffith functional. &nbsp The control of the crack front in the elastic body is performed by the optimum shape design technique. The Griffith functional is minimized with respect to the shape and the location of small inclusions in the body. The inclusions are located far from the crack. In order to minimize the Griffith functional over an admissible family of inclusions, the second order directional, mixed shape-topological derivatives of the elastic energy functional are evaluated. &nbsp The domain decomposition technique [42] is applied to the shape [56] and topological [54,55] sensitivity analysis of variational inequalities. &nbsp The nonlinear crack model in the framework of linear elasticity is considered in two and three spatial dimensions. The boundary value problem for the elastic displacement field takes the form of a variational inequality over the positive cone in a fractional Sobolev space. The variational inequality leads to a problem of metric projection over a polyhedric convex cone, so the concept of conical differentiability applies to shape and topological sensitivity analysis of variational inequalities under consideration.