Shape Optimization Formulation Research Articles

Automated shape and thickness optimization for non-matching isogeometric shells using free-form deformation

AbstractIsogeometric analysis (IGA) has emerged as a promising approach in the field of structural optimization, benefiting from the seamless integration between the computer-aided design (CAD) geometry and the analysis model by employing non-uniform rational B-splines (NURBS) as basis functions. However, structural optimization for real-world CAD geometries consisting of multiple non-matching NURBS patches remains a challenging task. In this work, we propose a unified formulation for shape and thickness optimization of separately parametrized shell structures by adopting the free-form deformation (FFD) technique, so that continuity with respect to design variables is preserved at patch intersections during optimization. Shell patches are modeled with isogeometric Kirchhoff–Love theory and coupled using a penalty-based method in the analysis. We use Lagrange extraction to link the control points associated with the B-spline FFD block and shell patches, and we perform IGA using the same extraction matrices by taking advantage of existing finite element assembly procedures in the FEniCS partial differential equation (PDE) solution library. Moreover, we enable automated analytical derivative computation by leveraging advanced code generation in FEniCS, thereby facilitating efficient gradient-based optimization algorithms. The framework is validated using a collection of benchmark problems, demonstrating its applications to shape and thickness optimization of aircraft wings with complex shell layouts.

Isogeometric Shape Optimization for Design Dependent Loads

Abstract This paper presents a new isogeometric formulation for shape optimization of structures subjected to design dependent loads. This work considers two types of design dependent loads, namely, surface loads like pressure where the direction and/or magnitude of force changes with the variation of boundary shape, and body forces that depend on the material layout. These problems have been mostly solved by topology optimization methods which are prone to difficulties in determination of the loading surface for pressure loads and problems associated with non-monotonous behavior of compliance and low density regions for body forces. This work uses an isogeometric shape optimization approach where the geometry is defined using non-uniform rational B-spline and the control point coordinates and control weights of the boundary are chosen as design variables. This approach accommodates the design dependent loads easily, in addition to its other advantages like exact geometry representation, local control, fewer design variables, excellent shape sensitivity, efficient mesh refinement strategies, and smooth results that can be integrated with computer aided design. Two classes of optimization problems have been discussed; they are minimum compliance problems subjected to volume constraint and minimum weight problems subjected to local stress constraints. These problems are solved using convex optimization programs. Hence, expressions for full sensitivities are derived which are new for structural shape optimization problems with design dependent loads. Some representative engineering examples are solved and compared with existing literature to demonstrate the application of the proposed method.

The continuous adjoint cut-cell method for shape optimization in cavitating flows

A continuous adjoint formulation for shape optimization in steady-state, cavitating flows is developed. A Transport Equation-based mixture model, extended with the Kunz cavitation model, is implemented to incorporate phase transition due to cavitation and for which the adjoint equations are derived. Flow and adjoint equations are discretized on 2D Cartesian meshes with cut-cells. In the cut-cell method, Cartesian cells intersected by the solid boundaries are reshaped by discarding their solid part, resulting in cells with an arbitrary number of sides. Emphasis is laid on the accurate computation of geometric sensitivities at the cut-cells, required by the adjoint method to compute the objective gradient. The sensitivity derivatives obtained via the adjoint method are compared with finite differences for the purpose of validation. The proposed adjoint formulation is assessed by performing three shape optimizations with two different objectives, namely cavitation suppression and hydrofoil lift maximization over isolated hydrofoils.

Shape optimization method for an inverse geometric source problem and stability at critical shape

<p style='text-indent:20px;'>This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.</p>

An improved shape optimization formulation of the Bernoulli problem by tracking the Neumann data

We propose a new shape optimization formulation of the Bernoulli problem by tracking the Neumann data. The associated state problem is an equivalent formulation of the Bernoulli problem with a Robin condition. We devise an iterative procedure based on a Lagrangian-like approach to numerically solve the minimization problem. The proposed scheme involves the knowledge of the shape gradient which is established through the minimax formulation. We illustrate the feasibility of the proposed method and highlight its advantage over the classical setting of tracking the Neumann data through several numerical examples.

A new energy-gap cost functional approach for the exterior Bernoulli free boundary problem

The solution to a free boundary problem of Bernoulli type, also known as Alt-Caffarelli problem, is studied via shape optimization techniques. In particular, a novel energy-gap cost functional approach with a state constraint consisting of a Robin condition is proposed as a shape optimization reformulation of the problem. Accordingly, the shape derivative of the cost is explicitly determined, and using the gradient information, a Lagrangian-like method is used to formulate an efficient boundary variation algorithm to numerically solve the minimization problem. The second order shape derivative of the cost is also computed, and through its characterization at the solution of the Bernoulli problem, the ill-posedness of the shape optimization formulation is proved. The analysis of the proposed formulation is completed by addressing the existence of optimal solution of the shape optimization problem and is accomplished by proving the continuity of the solution of the state problems with respect to the domain. The feasibility of the newly proposed method and its comparison with the classical energy-gap type cost functional approach is then presented through various numerical results. The numerical exploration issued in the study also includes results from a second-order optimization procedure based on a Newton-type method for resolving such minimization problem. This computational scheme put forward in the paper utilizes the Hessian information at the optimal solution and thus offers a state-of-the-art numerical approach for solving such free boundary problem via shape optimization setting.

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

AbstractIn 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)

Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty

We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded, for which the worst-case in terms of recirculation effects is inferred to correspond to a strong orifice flow through near-complete occlusion.A worst-case optimal control approach is applied to the steady Navier-Stokes equations in 2D to identify an anastomosis angle and a cuffed shape that are robust with respect to a possible range of residual flows. We also consider a reduced order modelling framework based on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model reduction or the robust framework.

On two different inverse form finding methods for hyperelastic and elastoplastic materials

AbstractThis contribution focuses on two different developments of mechanical‐computational methods for the optimal determination of the initial shape of formed functional components knowing the deformed configuration, the applied loads and the boundary conditions. The first method uses an inverse mechanical formulation and can be applied to materials with hyperelastic behaviors. For materials with elastoplastic properties this method is not advocated, without knowing the final plastic strains, due to the non uniqueness of the solution. The second method uses a shape optimization formulation in the sense of an inverse problem via successive iterations of the direct problem. For hyperelastic materials the inverse mechanical formulation is preferred for its velocity and the non exhibition of possible mesh distortions. In the shape optimization formulation mesh distortions can be avoided by an update of the reference configuration of the functional part. Both methods are using a formulation in the logarithmic strain space. A numerical example for materials with isotropic elastoplastic behaviors illustrates the shape optimization formulation. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computation of effective free surfaces in two phase flows

In this investigation we revisit the concept of “effective free surfaces” arising in the solution of the time-averaged fluid dynamics equations in the presence of free boundaries. This work is motivated by applications of the optimization and optimal control theory to problems involving free surfaces, where the time-dependent formulations lead to many technical difficulties which are, however, alleviated when steady governing equations are used instead. By introducing a number of precisely stated assumptions we develop and validate an approach in which the interface between the different phases, understood in the time-averaged sense, is sharp. In the proposed formulation, the terms representing the fluctuations of the free boundaries and of the hydrodynamic quantities appear as boundary conditions on the effective surface and require suitable closure models. As a simple model problem we consider impingement of free-falling droplets onto a fluid in a pool with a free surface, and a simple algebraic closure model is proposed for this system. The resulting averaged equations are of the free-boundary type and an efficient computational approach based on shape optimization formulation is developed for their solution. The computed effective surfaces exhibit consistent dependence on the problem parameters and compare favorably with the results obtained when the data from the actual time-dependent problem are used in lieu of the closure model.

Towards Inverse Form Finding Methods for a Deep Drawing Steel DC04

A challenge in the design of functional parts in metal forming processes is the determination of the initial, undeformed shape such that under a given load a part will obtain the desired deformed shape. An inverse mechanical or a shape optimization formulation might be used to solve this problem, which is inverse to the standard kinematic analysis in which the undeformed shape is known and the deformed shape unknown. The objective of the inverse mechanical formulation aims in the inverse deformation map that determines the (undeformed) material configuration, where the spatial (deformed) configuration and the mechanical loads are given. The shape optimization formulation predicts the initial shape in the sense of an inverse problem via successive iterations of the direct problem. In this paper, both methods are presented using a formulation in the logarithmic strain space. An update of the reference configuration of the sheet of metal during the optimization process is proposed in order to avoid mesh distortions. A first example showed the results obtained with both methods in isotropic hyperelasticity. A second example illustrated a simplified deep drawing computed with the shape optimization formulation in isotropic elastoplasticity. From the undeformed shapes obtained with both methods the deformed shapes are acquired with the direct mechanical formulation. Compared to the target deformed shape a minor difference in node coordinates is found. The computation time is lower with the inverse mechanical formulation in hyperelasticity. The update of the reference configuration in the shape optimization formulation allowed to avoid mesh distortions but increased the computational costs.

Optimal shape design of a floating body for minimal water wave forces

Water waves are the most significant excitation source for floating vessels. The motion of floating vessels needs to be stable against a random water wave environment. In this study, the authors try to find an optimal shape for a floating body that gives minimal water wave excitation forces for a given water wave over a predefined frequency band. First, we propose a shape optimization formulation with a displacement constraint of which the objective function is to minimize water–structure interaction forces. For the calculation of water wave forces, high-order boundary-element analysis software is developed under an assumption of linear hydrodynamic potential theory. The analysis software is plugged into an optimization toolbox in MATLAB software in order to find the optimum under-water shape of the floating body. Two numerical examples are introduced to show the validity of the proposed optimization formulation: hemisphere and pontoon problems. Various shapes that give minimal heave and/or surge water-excitation forces are sought.

A shape optimization formulation of weld pool determination

In this paper, we propose a shape optimization formulation for a problem modeling a process of welding. We show the existence of an optimal solution. The finite element method is used for the discretization of the problem. The discrete problem is solved by an identification technique using a parameterization of the weld pool by Bézier curves and Genetic algorithms.

On a Bernoulli problem with geometric constraints

A Bernoulli free boundary problem with geometrical constraints is studied. The domain $\Om$ is constrained to lie in the half space determined by $x_1\geq 0$ and its boundary to contain a segment of the hyperplane $\{x_1=0\}$ where non-homogeneous Dirichlet conditions are imposed. We are then looking for the solution of a partial differential equation satisfying a Dirichlet and a Neumann boundary condition simultaneously on the free boundary. The existence and uniqueness of a solution have already been addressed and this paper is devoted first to the study of geometric and asymptotic properties of the solution and then to the numerical treatment of the problem using a shape optimization formulation. The major difficulty and originality of this paper lies in the treatment of the geometric constraints.

Heat conduction in a solid slab embedded with a pipe of general cross-section: Shape Factor and Shape Optimization

We address the problem of two-dimensional heat conduction in a solid slab embedded with an isothermal, symmetric pipe of general cross-section. Similar formulations have applications in continuum mechanics and electricity. The main objective of this work is to develop a Shape Optimization algorithm that will reveal the optimal shapes of the pipe such that the conduction rate is maximized or minimized. This is achieved by optimizing the Shape Factor. To obtain the Shape Factor we transform the pipe into a strip using the generalized Schwarz-Christoffel transformation, and develop an integral equation of the first kind for the temperature gradient using Fourier transform techniques. The integral equation is solved both numerically and analytically/asymptotically. The fact that the Shape Factor is a monotonic function of the length of the strip suggests a Shape Optimization formulation where the objective function is the length of the strip and the variables of the optimization are the parameters of the generalized Schwarz–Christoffel transformation. Optimal shapes for the problem of minimizing the conduction rate are computed numerically and validated with an analytical solution. Numerical results for maximizing the transport rate are also obtained. The versatility and the robustness of the numerical optimization algorithm offers opportunities for improving the design of similar processes with non-linear equality and inequality constraints.

Shape optimization of a body located in low Reynolds number flow

AbstractThe purpose of this study is to perform a numerical application of the shape optimization formulation of a body located in an incompressible viscous flow field. The formulation is based on an optimal control theory in which a performance function of the fluid force is introduced. The performance function should be minimized satisfying the state equation. This problem can be transformed into the minimization problem without constraint condition by the Lagrange multiplier method and the adjoint equations using adjoint variables corresponding to the state equations. As a numerical study, the drag force minimization problem in the steady Stokes flow, which means approximated equation of the low Reynolds number Navier–Stokes equation is carried out. After that, the unsteady Navier–Stokes flow is analysed. As the minimization algorithm, the steepest descent method is successfully applied. Copyright © 2005 John Wiley & Sons, Ltd.

Remesh-free shape optimization using the wavelet-Galerkin method

The objective of this investigation is to develop a new remesh-free shape optimization method based on the adaptive wavelet-Galerkin analysis. To avoid cumbersome remeshing processes, we embed the original analysis design domain inside a simple fictitious domain and use the predetermined analysis grid points. For efficient adaptive analysis, we utilize the multiscale multiresolution characteristics of the hat interpolation wavelets that are associated with predetermined grid points. Furthermore, we approximate the design boundary curve by oblique lines that do not necessarily pass through the analysis grid points and develop an accurate stress evaluation technique based on the proposed boundary approximation. The validity of the proposed remesh-free shape optimization formulation based on the wavelet-Galerkin method is verified through benchmark shape optimization problems.

An Adjoint Formulation for Shape Optimization in Free-Surface Potential Flow

This paper is a step towards the application of automatic shape optimization techniques to the hydrodynamic design of marine vehicles. The flow is assumed to be irrotational and free-surface conditions are linearized. Several objective functionals of relevance to free-surface flow are proposed, and adjoint formulations for the determination of their gradients are derived. The accuracy of the gradient using the adjoint formulations has been verified by comparisons with direct calculations using second-order finite differences. The method is then used to obtain an optimal solution to the inverse problem for a three-dimensional body submerged near a free surface, and to design a minimum wave resistance body.

Meshless approach to shape optimization of linear thermoelastic solids

AbstractThis paper presents a formulation for shape optimization in thermoelasticity using a meshless method, namely the element‐free Galerkin method. Two examples are treated in detail and comparisons with previously published finite element analysis results demonstrate the excellent opportunities the EFG offers for solving these types of problems. Smoother stresses, no remeshing, and better accuracy than finite element solutions, permit answers to shape optimization problems in thermoelasticity that are practically unattainable with the classical FEM without remeshing. For the thermal fin example, the EFG finds finger shapes that are missed by the FEM analysis, and the objective value is greatly improved compared to the FEM solution. A study of the influence of the number of design parameters is performed and it is observed that the EFG can give better results with a smaller number of design parameters than is possible with traditional methods. Copyright © 2001 John Wiley & Sons, Ltd.

Boundary integral method for shape optimization of interface problems and its application to implant design in dentistry

A boundary integral formulation for shape optimization is presented in linear elasticity problems with zoned-inhomogeneous bodies. Sensitivity formula for arbitrary functional with respect to the interface shape variation is derived based on a boundary integral identity. Analytic as well as numerical examples for the sensitivity analysis are presented to show the validity of the formulation. An optimum design problem to find interface shape of endosseous implant in dentistry is studied under the objective of uniform stress at the interface between the implant and the bone. Though the problem involves numerous simplifications that may lead to the lack of reality, it is worthwhile to mention that the optimum shapes look similar to those of tooth roots. This may reflect a kind of natural optimum.