Numerical method | EPFL Graph Search

In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm. Mathematical definition Let F(x,y)=0 be a well-posed problem, i.e. F:X \times Y \rightarrow \mathbb{R} is a real or complex functional relationship, defined on the cross-product of an input data set X and an output data set Y, such that exists a locally lipschitz function g:X \rightarrow Y called resolvent, which has the property that for every root (x,y) of F, y=g(x). We define numerical method for the approximation of F(x,y)=0, the sequence of problems : \left { M_n \right }{n \in \mathbb{N}} = \left { F_n(x_n,y_n)=0 \right }{n \in \mathbb{N}}, with F_n:X_n \times Y_n \rightarrow \mathbb{R}

An octree-based adaptive semi-Lagrangian free surface flow solver

Viljami Henrikki Laurmaa

A numerical method based on an adaptive octree space discretization for the simulation of 3D free-surface fluid flows is proposed. The Navier-Stokes equations are solved with a time-splitting scheme, which decouples advection from diffusion/incompressibility. The advection step is solved with a semi-Lagrangian VOF-based scheme on the octree. An interface prediction algorithm is used to refine the octree at the predicted location of the interface in order to ensure detail preservation. Subsequently, the fluid is advected and a coarsening algorithm adapts the mesh to avoid excess refinement in non-interfacial regions. SLIC and decompression algorithms are used for post-processing to limit numerical diffusion and correct numerical compression of the VOF function. The octree scheme allows anisotropy, refinement of interfacial cells to an arbitrary level and supports arbitrary complex domains. It does not require a 2:1 cell size ratio condition between adjacent cells. The octree is then coupled with a tetrahedral mesh on which we solve the second step of the splitting algorithm, the Stokes' equations. Numerical validation is done on both advection benchmark test cases and results are compared with the uniform cell grid scheme. Paddle-generated water waves are also simulated and results are compared with experimental water wave profile measurements. \bigskip First order finite element stabilization schemes for the time-dependent Stokes' equations are studied. A unified proof of stability and convergence of velocity and pressure for consistent and non-consistent PSPG schemes for the time-dependent Stokes' equations is given with explicit dependence on viscosity and stabilization parameter. The link between bubble enrichment and Pressure Stabilized Petrov-Galerkin (PSPG) schemes in the context of time-dependent Stokes' equations is discussed and two bubble-based PSPG-type schemes are studied. Different possibilities for stabilization parameters are discussed. Numerical comparisons are done to determine stability, convergence and conditioning issues associated with different PSPG schemes, bubble-based schemes and local pressure projection schemes in different settings.

EPFL2016

High Performance Computing of flows around hydrofoils

Raimondo Pictet

Hydrocontest is a student competition involving teams from different universities around Switzerland and the world. This projects aims at providing an analysis of the performances of hydrofoils by means of computational fluid dynamics in a High Performance Computing context. The numerical scheme is based on the Finite Elements method for the spatial approximation of the Navier-Stokes equations, while on Backward Differentiation Formulas for the time discretization. The numerical implementation is performed in the parallel Finite Elements library LifeV.

2015

Optimal control of mass transfer in peritoneal dialysis

Diego Mastalli

The aim of this work is to set up mathematical models and numerical methods to investigate the mass transfer process occurring during the peritoneal dialysis (PD) therapy. More precisely the final goal is the set up of tools to find for each patient submitted to PD the best therapy profile to purify blood and remove water from the patients. First, we build up specific mathematical models to represent the physical phenomena we are interested in. As discussed in chapter 1, starting from the Kedem-Katchalsky equations, we end up with a systems of nonlinear ordinary differential equations describing the various aspects of the physical problem. Then, we propose a method based on nonlinear programming techniques to solve the inverse problem arising from the need to assess the peritoneal membrane characteristics which are not directly measurable on the patient. Thanks to its flexibility we are able to support the main standard tests nowadays in use to assess the kinetic properties of the peritoneum. We devise a suitable parametrization of the control function (DPD Dynamic Peritoneal Dialysis) that allows to improve the standard PD profiles with a larger set of treatments. Then we propose an optimization algorithm to improve the PD efficiency. Moreover, in the framework of control theory, we devise an algorithm based on the maximum principle of Pontryagin for switched systems to investigate deeply the PD optimal control problem. Afterwards we carry out numerical simulations, investigating the main inputs influencing the peritoneal dialysis efficiency. Specifically we show an extensive comparison between the APD (Automated Peritoneal Dialysis) and DPD (Dynamic Peritoneal Dilaysis) in order to assess the conditions under which DPD allows to improve the PD performance. Then a numerical investigation based on the algorithm devised for switched systems is carried out to assess the adequacy of DPD. Moreover we set up a procedure to reach an efficiency target and minimize the patient's exposure to glucose in order to improve the PD biocompatibility. Finally, we present the validation results of the mathematical model in order to verify its accuracy. A comparison between the APD and DPD is presented. Moreover we carry out a statistical analysis to assess the error distribution related to the most relevant quantities in order to evaluate strengths and weaknesses of this model and to identify the needs for a further improvement.

EPFL2006