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 general mathematical framework for constrained mixed-variable blackbox optimization problems with meta and categorical variables

A mathematical framework for modelling constrained mixed-variable optimization problems is presented in a blackbox optimization context. The framework introduces a new notation and allows solution strategies. The notation framework allows meta and categorical variables to be explicitly and efficiently modelled, which facilitates the solution of such problems. The new term meta variables is used to describe variables that influence which variables are acting or nonacting: meta variables may affect the number of variables and constraints. The flexibility of the solution strategies supports the main blackbox mixed-variable optimization approaches: direct search methods and surrogate-based methods (Bayesian optimization). The notation system and solution strategies are illustrated through an example of a hyperparameter optimization problem from the machine learning community

Derivative-free methods for mixed-integer nonsmooth constrained optimization

In this paper, we consider mixed-integer nonsmooth constrained optimization problems whose objective/constraint functions are available only as the output of a black-box zeroth-order oracle (i.e., an oracle that does not provide derivative information) and we propose a new derivative-free linesearch-based algorithmic framework to suitably handle those problems. We first describe a scheme for bound constrained problems that combines a dense sequence of directions (to handle the nonsmoothness of the objective function) with primitive directions (to handle discrete variables). Then, we embed an exact penalty approach in the scheme to suitably manage nonlinear (possibly nonsmooth) constraints. We analyze the global convergence properties of the proposed algorithms toward stationary points and we report the results of an extensive numerical experience on a set of mixed-integer test problems

Advances in the development of the discrete element method for excavation processes

This work presents new developments of the discrete element method improving e ciency and accuracy of modelling of rock-like materials, especially in excavation processes.Postprint (published version

Advances in the development of the discrete element method for excavation processes

Modelling of granular materials, soils and rocks has been a challenging topic of investigation for decades. Classical continuum mechanics has been used to idealize soils and rocks, and numerical solution techniques such as finite element method (FEM) has been used to model these materials. Considering the idealization of the material, continuum mechanics allows the analysis of phenomena with discontinuous nature such as fracture in rock or soil via damage models. However, in more complex processes like rock milling or crushing, this kind of models are usually not suitable. Discrete models are more appropriate for problems with multiple discontinuities and particulate materials. The discrete element method (DEM) has been gaining popularity in analysis of granular materials, soils and rocks. Many aspects of this method still require more profound investigation. This thesis presents new developments of the discrete element method improving effi ciency and accuracy of modelling of rock-like materials, especially in excavation processes. All the numerical algorithms has been implemented in an in-house software, which was then used to run numerical examples. The basic formulation of DEM with linear elastic-perfectly brittle contact model is presented. The main di erence with other models found in the literature is the consideration of global sti ness and strength parameters that are constants in the whole model. The result of a simulations is strongly related with the con guration of the particle assembly used. Particle assemblies should be su ciently compact and ensure the isotropy to reproduce the physical properties of the modelled material. This thesis presents a novel technique for the generation of highly dense particle assemblies in arbitrary geometries, satisfying all the requirements for accurate discrete element simulations. One of the key issues in the use of the DEM is the estimation of the contact model parameters. A methodology is proposed for the estimation of the contact model parameters yielding required macroscopic properties of the material. The relationships between the contact model parameters and the mechanical properties of brittle materials, as well as the influence of the particles assembly con guration on the macroscopic properties, are analysed. A major di culty in the application of the DEM to real engineering problems is the high computational cost in simulation involving a large number of particles. The most common way to solve this is the use of parallel computing techniques, where multiple processors are used. As an alternative, a coupling scheme between DEM and the finite element method (FEM) is proposed in the thesis. Within the hybrid DEM/FEM model, DEM is only used in the region of the domain where it provides an advantage over a continuum-based approach, as the FEM. The coupling is dynamically adapted, starting with the whole domain discretized with FEM. During the simulation, in the regions where a high stress level are found, a change of modelling method from continuum FEM to the discrete DEM is employed. Finally, all the developments are applied to the simulation of a real excavation process. An analysis of the rock cutting process with TBM disc cutters is performed, where DEM and the DEM/FEM coupling technique presents an important advantage over other simulation techniques.La modelación de materiales granulares, terrenos y rocas ha sido un desafío para la investigación por décadas. La mecánica del continuo clásica ha sido utilizada para idealizar terrenos y rocas, y técnicas numéricas de solución, como el método de los elementos finitos (FEM), han sido usadas para modelar estos materiales. Considerando la idealización del material, la mecánica del continuo permite el análisis de fenómenos de naturaleza discontinua como la fractura en rocas y terreno mediante modelos de daño. Sin embargo, en procesos mas complejos como la molienda o trituración de roca, este tipo de modelos no suelen ser adecuados. Los modelos discretos son mas apropiados para problemas con múltiples discontinuidades y material particulado. El método de los elementos discretos (DEM) ha ido ganando popularidad en el análisis de materiales granulares, terrenos y rocas. Sin embargo, muchos aspectos de este método todavía requieren una investigación mas profunda. Esta tesis presenta nuevos desarrollos del método de los elementos discretos para mejorar su eficiencia y precisión en el modelado de materiales como roca, especialmente para procesos de excavación. Todos los algoritmos numéricos se han implementado en el programa propio, que ha sido utilizado para probar distintos ejemplos. La formulación básica del DEM, con un modelo lineal de contacto elástico perfectamente frágil ha sido utilizado en el presente trabajo. La principal diferencia con otros modelos de la literatura es la consideración de que los parámetros de rigidez y fuerzas máximas son valores globales y constantes en todo el modelo. El resultado de la simulación está fuertemente relacionado con la configuración del ensamblaje de partículas utilizado. El ensamblaje de partículas debe ser suficientemente compacto y asegurar la isotropía de las propiedades físicas del material modelado. La tesis presenta una nueva técnica para la generación de ensamblajes de partículas de alta densidad para geometrías arbitrarias, satisfaciendo todos los requisitos para una simulación con elementos discretos correcta. Uno de los temas clave en el uso del DEM es la estimación de los parámetros del modelo de contacto. Se propone una metodología para la estimación de los parámetros del modelo de contacto siguiendo las propiedades macroscópicas requeridas en el material Las relaciones entre los parámetros del modelo y las propiedades mecánicas de materiales frágiles, así como su la influencia de la configuración del ensamblaje de partículas son analizadas. Una gran dificultad en la aplicación del DEM en problemas reales de ingeniería es el alto costo computacional de simulaciones que consideran un gran número de partículas. La solución mas común para resolver esto es el uso de técnicas de computación paralela, donde se utiliza un gran número de procesadores. Como vía alternativa, un esquema acoplado entre el DEM y el FEM expuesto en la tesis. Con el modelo híbrido DEM/FEM, el DEM es usado solo en las partes del dominio donde presenta ventajas sobre el enfoque continuo del FEM. El acoplamiento puede ser adaptado dinámicamente, comenzando con todo el dominio discretizado con FEM, y durante la simulación, en las regiones donde se encuentran altos niveles de tensión, se emplea un cambio del método de simulación de continuo (FEM) a discreto (DEM). Finalmente, todos los desarrollos son aplicados a la simulación de un proceso excavación real. Se realiza un estudio del proceso de corte de roca con discos costadores, utilizados en tuneladoras, donde el DEM y la técnica de acoplamiento presentan una importante ventaja sobre otras técnicas de simulación

Interface Tracking and Solid-Fluid Coupling Techniques with Coastal Engineering Applications

Multi-material physics arise in an innumerable amount of engineering problems. A broadly scoped numerical model is developed and described in this thesis to simulate the dynamic interaction of multi-fluid and solid systems. It is particularly aimed at modelling the interaction of two immiscible fluids with solid structures in a coastal engineering context; however it can be extended to other similar areas of research. The Navier Stokes equations governing the fluids are solved using a combination of finite element (FEM) and control volume finite element (CVFE) discretisations. The sharp interface between the fluids is obtained through the compressive transport of material properties (e.g. material concentration). This behaviour is achieved through the CVFE method and a conveniently limited flux calculation scheme based on the Hyper-C method by Leonard (1991). Analytical and validation test cases are provided, consisting of steady and unsteady flows. To further enhance the method, improve accuracy, and exploit Lagrangian benefits, a novel moving mesh method is also introduced and tested. It is essentially an Arbitrary Lagrangian Eulerian method in which the grid velocity is defined by semi-explicitly solving an iterative functional minimisation problem. A multi-phase approach is used to introduce solid structure modelling. In this approach, solution of the velocity field for the fluid phase is obtained using Model B as explained by Gidaspow (1994, page 151). Interaction between the fluid phase and the solids is achieved through the means of a source term included in the fluid momentum equations. The interacting force is calculated through integration of this source term and adding a buoyancy contribution. The resulting force is passed to an external solid-dynamics model such as the Discrete Element Method (DEM), or the combined Finite Discrete Element Method (FEMDEM). The versatility and novelty of this combined modelling approach stems from its ability to capture the fluid interaction with particles of random size and shape. Each of the three main components of this thesis: the advection scheme, the moving mesh method, and the solid interaction are individually validated, and examples of randomly shaped and sized particles are shown. To conclude the work, the methods are combined together in the context of coastal engineering applications, where the complex coupled problem of waves impacting on breakwater amour units is chosen to demonstrate the simulation possibilities. The three components developed in this thesis significantly extend the application range of already powerful tools, such as Fluidity, for fluids-modelling and finite discrete element solids-modelling tools by bringing them together for the first time

Simulating Fractures with Bonded Discrete Element Method

Along with motion and deformation, fracture is a fundamental behaviour for solid materials, playing a critical role in physically-based animation. Many simulation methods including both continuum and discrete approaches have been used by the graphics community to animate fractures for various materials. However, compared with motion and deformation, fracture remains a challenging task for simulation, because the material's geometry, topology and mechanical states all undergo continuous (and sometimes chaotic) changes as fragmentation develops. Recognizing the discontinuous nature of fragmentation, we propose a discrete approach, namely the Bonded Discrete Element Method (BDEM), for fracture simulation. The research of BDEM in engineering has been growing rapidly in recent years, while its potential in graphics has not been explored. We also introduce several novel changes to BDEM to make it more suitable for animation design. Compared with other fracture simulation methods, the BDEM has some attractive benefits, e.g. efficient handling of multiple fractures, simple formulation and implementation, and good scaling consistency. But it also has some critical weaknesses, e.g. high computational cost, which demand further research. A number of examples are presented to demonstrate the pros and cons, which are then highlighted in the conclusion and discussion

Parallel Multiscale Contact Dynamics for Rigid Non-spherical Bodies

The simulation of large numbers of rigid bodies of non-analytical shapes or vastly varying sizes which collide with each other is computationally challenging. The fundamental problem is the identification of all contact points between all particles at every time step. In the Discrete Element Method (DEM), this is particularly difficult for particles of arbitrary geometry that exhibit sharp features (e.g. rock granulates). While most codes avoid non-spherical or non-analytical shapes due to the computational complexity, we introduce an iterative-based contact detection method for triangulated geometries. The new method is an improvement over a naive brute force approach which checks all possible geometric constellations of contact and thus exhibits a lot of execution branching. Our iterative approach has limited branching and high floating point operations per processed byte. It thus is suitable for modern Single Instruction Multiple Data (SIMD) CPU hardware. As only the naive brute force approach is robust and always yields a correct solution, we propose a hybrid solution that combines the best of the two worlds to produce fast and robust contacts. In terms of the DEM workflow, we furthermore propose a multilevel tree-based data structure strategy that holds all particles in the domain on multiple scales in grids. Grids reduce the total computational complexity of the simulation. The data structure is combined with the DEM phases to form a single touch tree-based traversal that identifies both contact points between particle pairs and introduces concurrency to the system during particle comparisons in one multiscale grid sweep. Finally, a reluctant adaptivity variant is introduced which enables us to realise an improved time stepping scheme with larger time steps than standard adaptivity while we still minimise the grid administration overhead. Four different parallelisation strategies that exploit multicore architectures are discussed for the triad of methodological ingredients. Each parallelisation scheme exhibits unique behaviour depending on the grid and particle geometry at hand. The fusion of them into a task-based parallelisation workflow yields promising speedups. Our work shows that new computer architecture can push the boundary of DEM computability but this is only possible if the right data structures and algorithms are chosen