Fireshape: a shape optimization toolbox for Firedrake

We introduce Fireshape, an open-source and automated shape optimization toolbox for the finite element software Firedrake. Fireshape is based on the moving mesh method and allows users with minimal shape optimization knowledge to tackle with ease challenging shape optimization problems constrained to partial differential equations (PDEs).Comment: 17 pages, 13 figures, 6 listing

A shape optimization algorithm for interface identification allowing topological changes

In this work we investigate a combination of classical PDE constrained optimization methods and a rounding strategy based on shape optimization for the identification of interfaces. The goal is to identify radioactive regions in a groundwater flow represented by a control that is either active or inactive. We use a relaxation of the binary problem on a coarse grid as initial guess for the shape optimization with higher resolution. The result is a computationally cheap method that does not have to perform large shape deformations. We demonstrate that our algorithm is moreover able to change the topology of the initial guess

Shape optimisation with nearly conformal transformations

In shape optimisation it is desirable to obtain deformations of a given mesh without negative impact on the mesh quality. We propose a new algorithm using least square formulations of the Cauchy-Riemann equations. Our method allows to deform meshes in a nearly conformal way and thus approximately preserves the angles of triangles during the optimisation process. The performance of our methodology is shown by applying our method to some unconstrained shape functions and a constrained Stokes shape optimisation problem

Suitable Spaces for Shape Optimization

The differential-geometric structure of certain shape spaces is investigated and applied to the theory of shape optimization problems constrained by partial differential equations and variational inequalities. Furthermore, we define a diffeological structure on a new space of so-called $H^{1/2}$-shapes. This can be seen as a first step towards the formulation of optimization techniques on diffeological spaces. The $H^{1/2}$-shapes are a generalization of smooth shapes and arise naturally in shape optimization problems

First and Second Order Shape Optimization based on Restricted Mesh Deformations

We consider shape optimization problems subject to elliptic partial differential equations. In the context of the finite element method, the geometry to be optimized is represented by the computational mesh, and the optimization proceeds by repeatedly updating the mesh node positions. It is well known that such a procedure eventually may lead to a deterioration of mesh quality, or even an invalidation of the mesh, when interior nodes penetrate neighboring cells. We examine this phenomenon, which can be traced back to the ineptness of the discretized objective when considered over the space of mesh node positions. As a remedy, we propose a restriction in the admissible mesh deformations, inspired by the Hadamard structure theorem. First and second order methods are considered in this setting. Numerical results show that mesh degeneracy can be overcome, avoiding the need for remeshing or other strategies. FEniCS code for the proposed methods is available on GitHub

Gradient-based Constrained Optimization Using a Database of Linear Reduced-Order Models

A methodology grounded in model reduction is presented for accelerating the gradient-based solution of a family of linear or nonlinear constrained optimization problems where the constraints include at least one linear Partial Differential Equation (PDE). A key component of this methodology is the construction, during an offline phase, of a database of pointwise, linear, Projection-based Reduced-Order Models (PROM)s associated with a design parameter space and the linear PDE(s). A parameter sampling procedure based on an appropriate saturation assumption is proposed to maximize the efficiency of such a database of PROMs. A real-time method is also presented for interpolating at any queried but unsampled parameter vector in the design parameter space the relevant sensitivities of a PROM. The practical feasibility, computational advantages, and performance of the proposed methodology are demonstrated for several realistic, nonlinear, aerodynamic shape optimization problems governed by linear aeroelastic constraints

An optimization-based approach for high-order accurate discretization of conservation laws with discontinuous solutions

This work introduces a novel discontinuity-tracking framework for resolving discontinuous solutions of conservation laws with high-order numerical discretizations that support inter-element solution discontinuities, such as discontinuous Galerkin methods. The proposed method aims to align inter-element boundaries with discontinuities in the solution by deforming the computational mesh. A discontinuity-aligned mesh ensures the discontinuity is represented through inter-element jumps while smooth basis functions interior to elements are only used to approximate smooth regions of the solution, thereby avoiding Gibbs' phenomena that create well-known stability issues. Therefore, very coarse high-order discretizations accurately resolve the piecewise smooth solution throughout the domain, provided the discontinuity is tracked. Central to the proposed discontinuity-tracking framework is a discrete PDE-constrained optimization formulation that simultaneously aligns the computational mesh with discontinuities in the solution and solves the discretized conservation law on this mesh. The optimization objective is taken as a combination of the the deviation of the finite-dimensional solution from its element-wise average and a mesh distortion metric to simultaneously penalize Gibbs' phenomena and distorted meshes. We advocate a gradient-based, full space solver where the mesh and conservation law solution converge to their optimal values simultaneously and therefore never require the solution of the discrete conservation law on a non-aligned mesh. The merit of the proposed method is demonstrated on a number of one- and two-dimensional model problems including 2D supersonic flow around a bluff body. We demonstrate optimal $\mathcal{O}(h^{p+1})$ convergence rates in the $L^1$ norm for up to polynomial order $p=6$ and show that accurate solutions can be obtained on extremely coarse meshes.Comment: 40 pages, 23 figures, 1 tabl

A Two Stage CVT / Eikonal Convection Mesh Deformation Approach for Large Nodal Deformations

A two step mesh deformation approach for large nodal deformations, typically arising from non-parametric shape optimization, fluid-structure interaction or computer graphics, is considered. Two major difficulties, collapsed cells and an undesirable parameterization, are overcome by considering a special form of ray tracing paired with a centroid Voronoi reparameterization. The ray direction is computed by solving an Eikonal equation. With respect to the Hadamard form of the shape derivative, both steps are within the kernel of the objective and have no negative impact on the minimizer. The paper concludes with applications in 2D and 3D fluid dynamics and automatic code generation and manages to solve these problems without any remeshing. The methodology is available as a FEniCS shape optimization add-on at http://www.mathematik.uni-wuerzburg.de/~schmidt/femorph

Constrained particle-mesh projections in a hybridized discontinuous Galerkin framework with applications to advection-dominated flows

By combining concepts from particle-in-cell (PIC) and hybridized discontinuous Galerkin (HDG) methods, we present a particle-mesh scheme which allows for diffusion-free advection, satisfies mass and momentum conservation principles in a local sense, and allows the extension to high-order spatial accuracy. To achieve this, we propose a novel particle-mesh projection operator required for the exchange of information between the particles and the mesh. Key is to cast these projections as a PDE-constrained $\ell^2$-optimization problem to allow the advective field naturally located on Lagrangian particles to be expressed as a mesh quantity. By expressing the control variable in terms of single-valued functions at cell interfaces, this optimization problem seamlessly fits in a HDG framework. Owing to this framework, the resulting scheme can be implemented efficiently via static condensation. The performance of the scheme is demonstrated by means of various numerical examples for the linear advection-diffusion equation and the incompressible Navier-Stokes equations. The results show that optimal spatial accuracy can be achieved, and given the particular time-stepping strategy, second-order time accuracy is confirmed. The robustness of the scheme is illustrated by considering benchmarks for advection of discontinuous fields and the Taylor-Green vortex instability in the high Reynolds number regime

An Adjoint Method for a High-Order Discretization of Deforming Domain Conservation Laws for Optimization of Flow Problems

The fully discrete adjoint equations and the corresponding adjoint method are derived for a globally high- order accurate discretization of conservation laws on parametrized, deforming domains. The conservation law on the deforming domain is transformed into one on a fixed reference domain by the introduction of a time-dependent mapping that encapsulates the domain deformation and parametrization, resulting in an Arbitrary Lagrangian-Eulerian form of the governing equations. A high-order discontinuous Galerkin method is used to discretize the transformed equation in space and a high-order diagonally implicit Runge- Kutta scheme is used for the temporal discretization. Quantities of interest that take the form of space-time integrals are discretized in a solver-consistent manner. The corresponding fully discrete adjoint method is used to compute exact gradients of quantities of interest along the manifold of solutions of the fully discrete conservation law. The adjoint method is used to solve two optimal shape and control problems governed by the isentropic, compressible Navier-Stokes equations. The first optimization problem seeks the energetically optimal trajectory of a 2D airfoil given a required initial and final spatial position. The optimization solver, driven by gradients computed via the adjoint method, reduced the total energy required to complete the specified mission nearly an order of magnitude. The second optimization problem seeks the energetically optimal flapping motion and time-morphed geometry of a 2D airfoil given an equality constraint on the x-directed impulse generated on the airfoil. The optimization solver satisfied the impulse constraint to greater than 8 digits of accuracy and reduced the required energy between a factor of 2 and 10, depending on the value of the impulse constraint, as compared to the nominal configuration.Comment: 37 pages, 17 figures, 6 table