Numerical approximation of phase field based shape and topology optimization for fluids

We consider the problem of finding optimal shapes of fluid domains. The fluid obeys the Navier--Stokes equations. Inside a holdall container we use a phase field approach using diffuse interfaces to describe the domain of free flow. We formulate a corresponding optimization problem where flow outside the fluid domain is penalized. The resulting formulation of the shape optimization problem is shown to be well-posed, hence there exists a minimizer, and first order optimality conditions are derived. For the numerical realization we introduce a mass conserving gradient flow and obtain a Cahn--Hilliard type system, which is integrated numerically using the finite element method. An adaptive concept using reliable, residual based error estimation is exploited for the resolution of the spatial mesh. The overall concept is numerically investigated and comparison values are provided

Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities

We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property

Proceedings of the FEniCS Conference 2017

Proceedings of the FEniCS Conference 2017 that took place 12-14 June 2017 at the University of Luxembourg, Luxembourg

A contribution to the finite element analysis of high-speed compressible flows and aerodynamics shape optimization

This work covers a contribution to two most interesting research elds in aerodynamics, the fi nite element analysis of high-speed compressible flows (Part I) and aerodynamic shape optimization (Part II). The fi rst part of this study aims at the development of a new stabilization formulation based on the Finite Increment Calculus (FIC) scheme for the Euler and Navier-Stokes equations in the context of the Galerkin nite element method (FEM). The FIC method is based on expressing the balance of fluxes in a spacetime domain of nite size. It is tried to prevent the creation of instabilities normally presented in the numerical solutions due to the high convective term and sharp gradients. In order to overcome the typical instabilities happening in the numerical solution of the high-speed compressible flows, two stabilization terms, called streamline term and transverse term, are added through the FIC formulation in space-time domain to the original conservative equations of mass, momentum and energy. Generally, the streamline term holding the direction of the velocity is responsible for stabilizing the spurious solutions produced from the convective term while the transverse term smooths the solution in the high gradient zones. An explicit fourth order Runge-Kutta scheme is implemented to advance the solution in time. In order to investigate the capability of the proposed formulation, some numerical test examples corresponding to subsonic, transonic and supersonic regimes for inviscid and viscous flows are presented. The behavior of the proposed stabilization technique in providing appropriate solutions has been studied especially near the zones where the solution has some complexities such as shock waves, boundary layer, stagnation point, etc. Although the derived methodology delivers precise results with a nearly coarse mesh, the mesh refinement technique is coupled in the solution to create a suitable mesh particularly in the high gradient zones. The comparison of the numerical results obtained from the FIC formulation with the reference ones demonstrates the robustness of the proposed method for stabilization of the Euler and Navier-Stokes equations. It is observed that the usual oscillations occur in the Galerkin FEM, especially near the high gradient zones, are cured by implementing the proposed stabilization terms. Furthermore, allowing the adaptation framework to modify the mesh, the quality of the results improves signi cantly. The second part of this thesis proposes a procedure for aerodynamic shape optimization combining Genetic Algorithm (GA) and mesh re nement technique. In particular, it is investigated the e ect of mesh re nement on the computational cost and solution accuracy during the process of aerodynamic shape optimization. Therefore, an adaptive remeshing technique is joined to the CFD solver for the analysis of each design candidate to guarantee the production of more realistic solutions during the optimum design process in the presence of shock waves. In this study, some practical transonic airfoil design problems using adap- tive mesh techniques coupled to Multi-Objective Genetic Algorithms (MOGAs) and Euler flow analyzer are addressed. The methodology is implemented to solve three practical design problems; the fi rst test case considers a reconstruction design optimization that minimizes the pressure error between a prede ned pressure curve and candidate pressure distribution. The second test considers the total drag minimization by designing airfoil shape operating at transonic speeds. For the final test case, a multi-objective design optimization is conducted to maximize both the lift to drag ratio (L/D) and lift coe cient (Cl). The solutions obtained with and without adaptive mesh re nement are compared in terms of solution accuracy and computational cost. These design problems under transonic speeds need to be solved with a ne mesh, particularly near the object, to capture the shock waves that will cost high computational time and require solution accuracy. By comparison of the the numerical results obtained with both optimization problems, the obtainment of direct bene ts in the reduction of the total computational cost through a better convergence to the final solution is evaluated. Indeed, the improvement of the solution quality when an adaptive remeshing technique is coupled with the optimum design strategy can be judged.El presente trabajo pretende contribuir a dos de los campos de investigaci on m as interesantes en la aerodin amica, el an alisis num erico de flujos compresibles a alta velocidad (Parte I) y la optimizaci on de la forma aerodin amica (Parte II). La primera parte de este estudio se centra en la soluci on num erica de las ecuaciones de Navier-Stokes, que modelan el comportamiento de flujos compresibles a alta velocidad. La discretizaci on espacial se lleva a cabo mediante el m etodo de elementos nitos (FEM) y se pone especial enfasis en el desarrollo de una nueva formulaci on estabilizada basada en la t ecnica de c alculo de Incremento fi nitos (FIC). En este ultima, los t erminos de estabilizaci on convectiva se obtienen de manera natural de las ecuaciones de gobierno a trav es de postulados de conservaci on y equilibrio de flujos en un dominio espacio-tiempo de tamaño nito. Ello lleva a la obtenci on de dos t erminos de estabilizaci on que funcionan de manera complementaria. Uno act ua en direcci on de las lineas de corriente proporcionando la estabilizaci on necesaria para contrarestrar las inestabilidades propias de la forma discreta de Galerkin y el otro t ermino, de tipo shock capturing, act ua de manera transversal a las l neas de corriente y permite mejorar la soluci on num erica alrededor de discontinuidades y otro tipos de fen omenos localizados en el campo de soluci on de problema. La forma discreta de las ecuaciones de gobierno se completa mediante un esquema de integraci on temporal expl icito de tipo de Runge-Kutta de 4to orden. El esquema de soluci on b asico propuesto se complementa con una t ecnica de re namiento adaptativo de malla que permite mejorar autom aticamente la soluci on num erica en zonas localizadas del dominio en que, dadas las caracter sticas del flujo, se necesita una mayor resoluci on espacial. Con el prop osito de investigar el comportamiento de la formulaci on num erica se estudian diferentes casos de an alisis que implican flujos viscosos y no viscosos en r egimen subs onico, trans onico y supers onico y se estudia con especial detalle el funcionamiento de la t ecnica de estabilizaci on propuesta. Los resultados obtenidos demuestran una exactitud satisfactoria y una buena correlaci on con resultados presentes en la literatura, incluso cuando se trabaja con discretizaciones espaciales relativamente gruesas. Adicionalmente, los estudios num ericos realizados demuestran que el empleo del esquema adaptativo de malla es e ficaz para incrementar la exactitud de la soluci on numerica manteniendo un bajo coste computacional. En la segunda parte de este estudio se propone un m etodo para la optimizaci on de formas aerodin amicas que combina algoritmos gen eticos multiobjetivo (MOGAs) y remallado adaptativo con el objetivo de asegurar, con un coste computacional m nimo, la calidad de la soluci on numerica empleada en el proceso de b usqueda de un determinado diseño objetivo, particularmente cuando el flujo presenta discontinuidades y gradientes muy localizados, ti picos de flujos a alta velocidad. La metodolog a se aplica a resolver tres problemas pr acticos de diseño de per les aerodin amicos en flujo trans onico que implican la optimizaci on de la distribuci on de presiones, minimizaci on de la resistencia de onda y maximizaci on conjunta de la sustentaci on y la relaci on sustentaci on/resistencia. Para cada uno de ellos se estudia el efecto del re namiento en la calidad de la soluci on num erica as como tambi en en el coste computacional y la convergencia del problema. Los estudios realizados demuestran la e cacia de la metodolog a propuesta

Application of general semi-infinite Programming to Lapidary Cutting Problems

We consider a volume maximization problem arising in gemstone cutting industry. The problem is formulated as a general semi-infinite program (GSIP) and solved using an interiorpoint method developed by Stein. It is shown, that the convexity assumption needed for the convergence of the algorithm can be satisfied by appropriate modelling. Clustering techniques are used to reduce the number of container constraints, which is necessary to make the subproblems practically tractable. An iterative process consisting of GSIP optimization and adaptive refinement steps is then employed to obtain an optimal solution which is also feasible for the original problem. Some numerical results based on realworld data are also presented

Multiphase flow of immiscible fluids on unstructured moving meshes

Figure 1: Multiple fluids with different viscosity coefficients and surface tension densities splashing on the bottom of a cylindrical container. Observe that the simulation has no problem dealing with thin sheets. In this paper, we present a method for animating multiphase flow of immiscible fluids using unstructured moving meshes. Our underlying discretization is an unstructured tetrahedral mesh, the deformable simplicial complex (DSC), that moves with the flow in a Lagrangian manner. Mesh optimization operations improve element quality and avoid element inversion. In the context of multiphase flow, we guarantee that every element is occupied by a single fluid and, consequently, the interface between fluids is represented by a set of faces in the simplicial complex. This approach ensures that the underlying discretization matches the physics and avoids the additional book-keeping required in grid-based methods where multiple fluids may occupy the same cell. Our Lagrangian approach naturally leads us to adopt a finite element approach to simulation, in contrast to the finite volume approaches adopted by a majority of fluid simulation techniques that use tetrahedral meshes. We characterize fluid simulation as an optimization problem allowing for full coupling of the pressure and velocity fields and the incorporation of a second-order surface energy. We introduce a preconditioner based on the diagonal Schur complement and solve our optimization on the GPU. We provide the results of parameter studies as well as