Alain Dervieux

Unstructured multigridding by volume agglomeration: Current status

Abstract We describe a multigrid (MG) method for solving the Euler equations as applied to non-structured meshes in two (triangles) and three dimensions (tetrahedra). The main idea is to coarsen the given mesh by using topological neighboring relations. It is applied to upwind solvers relying on the MUSCL methodology. Two MG schemes are presented: an explicit Runge-Kutta FAS, and an implicit correction scheme. Transonic external flow computations are described for illustration.

Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes

In this paper, certain well-known upwind schemes for hyperbolic equations are extended to solve the two-dimensional Saint-Venant (or shallow water) equations. We consider unstructured meshes and a new type of finite volume to obtain a suitable treatment of the boundary conditions. The source term involving the gradient of the depth is upwinded in a similar way as the flux terms. The resulting schemes are compared in terms of a conservation property. For the time discretization we consider both explicit and implicit schemes. Finally, we present the numerical results for tidal flows in the Pontevedra ria, Galicia, Spain.

Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations

This paper studies the coupling between anisotropic mesh adaptation and goal-oriented error estimate. The former is very well suited to the control of the interpolation error. It is generally interpreted as a local geometric error estimate. On the contrary, the latter is preferred when studying approximation errors for PDEs. It generally involves non local error contributions. Consequently, a full and strong coupling between both is hard to achieve due to this apparent incompatibility. This paper shows how to achieve this coupling in three steps. First, a new a priori error estimate is proved in a formal framework adapted to goal-oriented mesh adaptation for output functionals. This estimate is based on a careful analysis of the contributions of the implicit error and of the interpolation error. Second, the error estimate is applied to the set of steady compressible Euler equations which are solved by a stabilized Galerkin finite element discretization. A goal-oriented error estimation is derived. It involves the interpolation error of the Euler fluxes weighted by the gradient of the adjoint state associated with the observed functional. Third, rewritten in the continuous mesh framework, the previous estimate is minimized on the set of continuous meshes thanks to a calculus of variations. The optimal continuous mesh is then derived analytically. Thus, it can be used as a metric tensor field to drive the mesh adaptation. From a numerical point of view, this method is completely automatic, intrinsically anisotropic, and does not depend on any a priori choice of variables to perform the adaptation. 3D examples of steady flows around supersonic and transsonic jets are presented to validate the current approach and to demonstrate its efficiency.

Exact-gradient shape optimization of a 2-D Euler flow

Abstract The optimization of an obstacle shape immersed in an Eulerian flow is investigated. In order to construct a descent method, we consider the differentiation of the flow solution with respect to the shape. In the continous case, the Hadamard variational formula yields the formal derivatives. In the discrete case, we choose an upwind method with flux splitting, and proved that an exact gradient can be derived using the adjoint state. The behavior of a gradient method is studied for a family of nozzle flows.

Mixed-element-volume MUSCL methods with weak viscosity for steady and unsteady flow calculations

Abstract One of the most desirable approximation property in compressible flow simulation is the avoidance of superfluous numerical spurious diffusion. We propose to reach this goal on unstructured meshes by the application of sophisticated upwind finite volume schemes involving sixth-order viscosity. The result is a family of second-order accurate schemes which exhibits much less diffusion than usual upwind finite volumes when applied to steady and unsteady flow simulations. The new schemes allow to compute accurately a difficult application such as a vortex–acoustic coupling in a rocket chamber.

Achievement of Global Second Order Mesh Convergence for Discontinuous Flows with Adapted Unstructured Meshes

In the context of steady CFD computations, some numerical experiments point out that only a global mesh convergence order of one is numerically reached on a sequence of uniformly refined meshes although the considered numerical scheme is second order. This is due to the presence of genuine discontinuities or sharp gradients in the modelled flow. In order to address this issue, a continuous mesh adaptation framework is proposed based on the metric notion. It relies on a L control of the interpolation error for twice differentiable functions. This theory provides an optimal bound of the interpolation error involving the Hessian of the solution. From this estimate, an optimal metric is exhibited to govern the adapted mesh generation. As regards steady flow computations with discontinuities, a global second order mesh convergence should be obtained. To this end, a higher order smooth approximation of the solution is reconstructed providing an accurate and reliable Hessian evaluation. Several numerical examples in two and three dimensions illustrate that the global convergence order is recovered using this mesh adaptation strategy.

A hierarchical approach for shape optimization

We consider the gradient method applied to the optimal control of a system for which each simulation is expensive. A method for increasing the number of unknowns, and relying on multilevel ideas is tested for the academic problem of shape optimization of a nozzle in a subsonic or transonic Euler flow.

Reverse Automatic Differentiation for Optimum Design: From Adjoint State Assembly to Gradient Computation

Gradient descent is a key technique in Optimal Design problems. We describe a method to compute the gradient of a optimization criterion with respect to design parameters. This method is hybrid, using Automatic Differentiation to compute the residual of the adjoint system, and using this residual in a hand-written solver that computes the adjoint state and then the gradient. Automatic Differentiation is here used in its so-called reverse mode, with a special refinement for gather-scatter loops. The hand-written solver uses a matrix-free algorithm, preconditioned by the first-order derivative of the flux function. This method was tested on a typical optimal design problem, for which we give validation and performance results.

Time accurate anisotropic goal-oriented mesh adaptation for unsteady flows

We present a new algorithm for combining an anisotropic goal-oriented error estimate with the mesh adaptation fixed point method for unsteady problems. The minimization of the error on a functional provides both the density and the anisotropy (stretching) of the optimal mesh. They are expressed in terms of state and adjoint. This method is used for specifying the mesh for a time sub-interval. A global fixed point iterates the re-evaluation of meshes and states over the whole time interval until convergence of the space-time mesh. Applications to unsteady blast-wave and acoustic-wave Euler flows are presented.

Multi-Dimensional Continuous Metric for Mesh Adaptation

Mesh adaptation is considered here as the research of an optimum that minimizes the P1 interpolation error of a function u of R given a number of vertices. A continuous modeling is described by considering classes of equivalence between meshes which are analytically represented by a metric tensor field. Continuous metrics are exhibited for L error model and mesh order of convergence are analyzed. Numerical examples are provided in two and three dimensions.