Quasi second-order methods for PDE-constrained forward and inverse problems

La conception assistée par ordinateur (CAO), les effets visuels, la robotique et de nombreux autres domaines tels que la biologie computationnelle, le génie aérospatial, etc. reposent sur la résolution de problèmes mathématiques. Dans la plupart des cas, des méthodes de calcul sont utilisées pour résoudre ces problèmes. Le choix et la construction de la méthode de calcul ont un impact important sur les résultats et l'efficacité du calcul. La structure du problème peut être utilisée pour créer des méthodes, qui sont plus rapides et produisent des résultats qualitativement meilleurs que les méthodes qui n'utilisent pas la structure. Cette thèse présente trois articles avec trois nouvelles méthodes de calcul s'attaquant à des problèmes de simulation et d'optimisation contraints par des équations aux dérivées partielles (EDP). Dans le premier article, nous abordons le problème de la dissipation d'énergie des solveurs fluides courants dans les effets visuels. Les solveurs de fluides sont omniprésents dans la création d'effets dans les courts et longs métrages d'animation. Nous présentons un schéma d'intégration temporelle pour la dynamique des fluides incompressibles qui préserve mieux l'énergie comparé aux nombreuses méthodes précédentes. La méthode présentée présente une faible surcharge et peut être intégrée à un large éventail de méthodes existantes. L'amélioration de la conservation de l'énergie permet la création d'animations nettement plus dynamiques. Nous abordons ensuite la conception computationelle dont le but est d'exploiter l'outils computationnel dans le but d'améliorer le processus de conception. Plus précisément, nous examinons l'analyse de sensibilité, qui calcule les sensibilités du résultat de la simulation par rapport aux paramètres de conception afin d'optimiser automatiquement la conception. Dans ce contexte, nous présentons une méthode efficace de calcul de la direction de recherche de Gauss-Newton, en tirant parti des solveurs linéaires directs épars modernes. Notre méthode réduit considérablement le coût de calcul du processus d'optimisation pour une certaine classe de problèmes de conception inverse. Enfin, nous examinons l'optimisation de la topologie à l'aide de techniques d'apprentissage automatique. Nous posons deux questions : Pouvons-nous faire de l'optimisation topologique sans maillage et pouvons-nous apprendre un espace de solutions d'optimisation topologique. Nous appliquons des représentations neuronales implicites et obtenons des résultats structurellement sensibles pour l'optimisation topologique sans maillage en guidant le réseau neuronal pendant le processus d'optimisation et en adaptant les méthodes d'optimisation topologique par éléments finis. Notre méthode produit une représentation continue du champ de densité. De plus, nous présentons des espaces de solution appris en utilisant la représentation neuronale implicite.Computer-aided design (CAD), visual effects, robotics and many other fields such as computational biology, aerospace engineering etc. rely on the solution of mathematical problems. In most cases, computational methods are used to solve these problems. The choice and construction of the computational method has large impact on the results and the computational efficiency. The structure of the problem can be used to create methods, that are faster and produce qualitatively better results than methods that do not use the structure. This thesis presents three articles with three new computational methods tackling partial differential equation (PDE) constrained simulation and optimization problems. In the first article, we tackle the problem of energy dissipation of common fluid solvers in visual effects. Fluid solvers are ubiquitously used to create effects in animated shorts and feature films. We present a time integration scheme for incompressible fluid dynamics which preserves energy better than many previous methods. The presented method has low overhead and can be integrated into a wide range of existing methods. The improved energy conservation leads to noticeably more dynamic animations. We then move on to computational design whose goal is to harnesses computational techniques for the design process. Specifically, we look at sensitivity analysis, which computes the sensitivities of the simulation result with respect to the design parameters to automatically optimize the design. In this context, we present an efficient way to compute the Gauss-Newton search direction, leveraging modern sparse direct linear solvers. Our method reduces the computational cost of the optimization process greatly for a certain class of inverse design problems. Finally, we look at topology optimization using machine learning techniques. We ask two questions: Can we do mesh-free topology optimization and can we learn a space of topology optimization solutions. We apply implicit neural representations and obtain structurally sensible results for mesh-free topology optimization by guiding the neural network during optimization process and adapting methods from finite element based topology optimization. Our method produces a continuous representation of the density field. Additionally, we present learned solution spaces using the implicit neural representation

Turbulent Details Simulation for SPH Fluids via Vorticity Refinement

A major issue in Smoothed Particle Hydrodynamics (SPH) approaches is the numerical dissipation during the projection process, especially under coarse discretizations. High-frequency details, such as turbulence and vortices, are smoothed out, leading to unrealistic results. To address this issue, we introduce a Vorticity Refinement (VR) solver for SPH fluids with negligible computational overhead. In this method, the numerical dissipation of the vorticity field is recovered by the difference between the theoretical and the actual vorticity, so as to enhance turbulence details. Instead of solving the Biot-Savart integrals, a stream function, which is easier and more efficient to solve, is used to relate the vorticity field to the velocity field. We obtain turbulence effects of different intensity levels by changing an adjustable parameter. Since the vorticity field is enhanced according to the curl field, our method can not only amplify existing vortices, but also capture additional turbulence. Our VR solver is straightforward to implement and can be easily integrated into existing SPH methods

Shallow waters simulation

Dissertação de mestrado integrado em Informatics EngineeringRealistic simulation and rendering of water in real-time is a challenge within the field of computer graphics, as it is very computationally demanding. A common simulation approach is to reduce the problem from 3D to 2D by treating the water surface as a 2D heightfield. When simulating 2D fluids, the Shallow Water Equations (SWE) are often employed, which work under the assumption that the water’s horizontal scale is much greater than it’s vertical scale. There are several methods that have been developed or adapted to model the SWE, each with its own advantages and disadvantages. A common solution is to use grid-based methods where there is the classic approach of solving the equations in a grid, but also the Lattice-Boltzmann Method (LBM) which originated from the field of statistical physics. Particle based methods have also been used for modeling the SWE, namely as a variation of the popular Smoothed-Particle Hydrodynamics (SPH) method. This thesis presents an implementation for real-time simulation and rendering of a heightfield surface water volume. The water’s behavior is modeled by a grid-based SWE scheme with an efficient single kernel compute shader implementation. When it comes to visualizing the water volume created by the simulation, there are a variety of effects that can contribute to its realism and provide visual cues for its motion. In particular, When considering shallow water, there are certain features that can be highlighted, such as the refraction of the ground below and corresponding light attenuation, and the caustics patterns projected on it. Using the state produced by the simulation, a water surface mesh is rendered, where set of visual effects are explored. First, the water’s color is defined as a combination of reflected and transmitted light, while using a Cook- Torrance Bidirectional Reflectance Distribution Function (BRDF) to describe the Sun’s reflection. These results are then enhanced by data from a separate pass which provides caustics patterns and improved attenuation computations. Lastly, small-scale details are added to the surface by applying a normal map generated using noise. As part of the work, a thorough evaluation of the developed application is performed, providing a showcase of the results, insight into some of the parameters and options, and performance benchmarks.Simulação e renderização realista de água em tempo real é um desafio dentro do campo de computação gráfica, visto que é muito computacionalmente exigente. Uma abordagem comum de simulação é de reduzir o problema de 3D para 2D ao tratar a superfície da água como um campo de alturas 2D. Ao simular fluidos em 2D, é frequente usar as equações de águas rasas, que funcionam sobre o pressuposto de que a escala horizontal da água é muito maior que a sua escala vertical. Há vários métodos que foram desenvolvidos ou adaptados para modelar as equações de águas rasas, cada uma com as suas vantagens e desvantagens. Uma solução comum é utilizar métodos baseados em grelhas onde existe a abordagem clássica de resolver as equações numa grelha, mas também existe o método de Lattice Boltzmann que originou do campo de física estatística. Métodos baseados em partículas também já foram usados para modelar as equações de águas rasas, nomeadamente como uma variação do popular método de SPH. Esta tese apresenta uma implementação para simulação e renderização em tempo real de um volume de água com uma superfície de campo de alturas. O comportamento da água é modelado por um esquema de equações de águas rasas baseado na grelha com uma implementação eficiente de um único kernel de compute shader. No que toca a visualizar o volume de água criado pela simulação, existe uma variedade de efeitos que podem contribuir para o seu realismo e fornecer dicas visuais sobre o seu movimento. Ao considerar águas rasas, existem certas características que podem ser destacadas, como a refração do terreno por baixo e correspondente atenuação da luz, e padrões de cáusticas projetados nele. Usando o estado produzido pela simulação, uma malha da superfície da água é renderizada, onde um conjunto de efeitos visuais são explorados. Em primeiro lugar, a cor da água é definida como uma combinação de luz refletida e transmitida, sendo que uma BRDF de Cook-Torrance é usada para descrever a reflexão do Sol. Estes resultados são depois complementados com dados gerados num passo separado que fornece padrões de cáusticas e melhora as computações de atenuação. Por fim, detalhes de pequena escala são adicionados à superfície ao aplicar um mapa de normais gerado com ruído. Como parte do trabalho desenvolvido, é feita uma avaliação detalhada da aplicação desenvolvida, onde é apresentada uma demonstração dos resultados, comentários sobre alguns dos parâmetros e opções, e referências de desempenho

Implicit High-Order Flux Reconstruction Solver for High-Speed Compressible Flows

The present paper addresses the development and implementation of the first high-order Flux Reconstruction (FR) solver for high-speed flows within the open-source COOLFluiD (Computational Object-Oriented Libraries for Fluid Dynamics) platform. The resulting solver is fully implicit and able to simulate compressible flow problems governed by either the Euler or the Navier-Stokes equations in two and three dimensions. Furthermore, it can run in parallel on multiple CPU-cores and is designed to handle unstructured grids consisting of both straight and curved edged quadrilateral or hexahedral elements. While most of the implementation relies on state-of-the-art FR algorithms, an improved and more case-independent shock capturing scheme has been developed in order to tackle the first viscous hypersonic simulations using the FR method. Extensive verification of the FR solver has been performed through the use of reproducible benchmark test cases with flow speeds ranging from subsonic to hypersonic, up to Mach 17.6. The obtained results have been favorably compared to those available in literature. Furthermore, so-called super-accuracy is retrieved for certain cases when solving the Euler equations. The strengths of the FR solver in terms of computational accuracy per degree of freedom are also illustrated. Finally, the influence of the characterizing parameters of the FR method as well as the the influence of the novel shock capturing scheme on the accuracy of the developed solver is discussed

Kinetic Solvers with Adaptive Mesh in Phase Space

An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for solving multi-dimensional kinetic equations by the discrete velocity method. A Cartesian mesh for both configuration (r) and velocity (v) spaces is produced using a tree of trees data structure. The mesh in r-space is automatically generated around embedded boundaries and dynamically adapted to local solution properties. The mesh in v-space is created on-the-fly for each cell in r-space. Mappings between neighboring v-space trees implemented for the advection operator in configuration space. We have developed new algorithms for solving the full Boltzmann and linear Boltzmann equations with AMPS. Several recent innovations were used to calculate the discrete Boltzmann collision integral with dynamically adaptive mesh in velocity space: importance sampling, multi-point projection method, and the variance reduction method. We have developed an efficient algorithm for calculating the linear Boltzmann collision integral for elastic and inelastic collisions in a Lorentz gas. New AMPS technique has been demonstrated for simulations of hypersonic rarefied gas flows, ion and electron kinetics in weakly ionized plasma, radiation and light particle transport through thin films, and electron streaming in semiconductors. We have shown that AMPS allows minimizing the number of cells in phase space to reduce computational cost and memory usage for solving challenging kinetic problems

A momentum-conserving, consistent, Volume-of-Fluid method for incompressible flow on staggered grids

The computation of flows with large density contrasts is notoriously difficult. To alleviate the difficulty we consider a consistent mass and momentum-conserving discretization of the Navier-Stokes equation. Incompressible flow with capillary forces is modelled and the discretization is performed on a staggered grid of Marker and Cell type. The Volume-of-Fluid method is used to track the interface and a Height-Function method is used to compute surface tension. The advection of the volume fraction is performed using either the Lagrangian-Explicit / CIAM (Calcul d'Interface Affine par Morceaux) method or the Weymouth and Yue (WY) Eulerian-Implicit method. The WY method conserves fluid mass to machine accuracy provided incompressiblity is satisfied which leads to a method that is both momentum and mass-conserving. To improve the stability of these methods momentum fluxes are advected in a manner "consistent" with the volume-fraction fluxes, that is a discontinuity of the momentum is advected at the same speed as a discontinuity of the density. To find the density on the staggered cells on which the velocity is centered, an auxiliary reconstruction of the density is performed. The method is tested for a droplet without surface tension in uniform flow, for a droplet suddenly accelerated in a carrying gas at rest at very large density ratio without viscosity or surface tension, for the Kelvin-Helmholtz instability, for a falling raindrop and for an atomizing flow in air-water conditions