Parametrization (geometry) | EPFL Graph Search

In mathematics, and more specifically in geometry, parametrization (or parameterization; also parameterisation, parametrisation) is the process of finding parametric equations of a curve, a surface, or, more generally, a manifold or a variety, defined by an implicit equation. The inverse process is called implicitization. "To parameterize" by itself means "to express in terms of parameters". Parametrization is a mathematical process consisting of expressing the state of a system, process or model as a function of some independent quantities called parameters. The state of the system is generally determined by a finite set of coordinates, and the parametrization thus consists of one function of several real variables for each coordinate. The number of parameters is the number of degrees of freedom of the system. For example, the position of a point that moves on a curve in three-dimensional space is determined by the time needed to reach the point when starting from a fixed origin.

Surface parameterization and optimum design methodology for hydraulic turbines

This thesis presents a methodology for the design optimization of hydraulic runner blades. The originality of the methodology comes from the geometric definition of the blade shapes, which uses parametric surfaces instead of a set of profiles. The main advantage of using surfaces is the number of parameters required. The use of surfaces requires a different technique for the blade construction when compared to traditional approaches. NURBS surfaces are used for the geometric representation, the properties provided by such parametric formulation permit to reach the necessary flexibility and accuracy to be attained. Moreover, the surface approach can be seen as a way to liberate the blade design from traditional discrete sectional approaches. Actual blade optimization procedures are a compromise between the quality of the design, its performance analysis and the subsequent computational time-consumed effort. The improvements provided by the above mentioned geometric definition allow the use of more realistic analysis tools for the design evaluation. Thus, Navier-Stokes (k – ε) simulations are integrated in a simple and direct optimization process. The resulting methodology is not penalized by the time-effort required and becomes of interest for its use in industrial applications. Finally, we supply a number of examples to demonstrate the feasibility of the optimization proposal. These examples illustrate the application of the methodology at different levels of geometric complexity. They are interesting not only through the results obtained, but also because they become acceptable in terms of time consumed on daily and industrial applications.

EPFL2006

Reduced Basis method for Isogeometric Analysis: application to structural problems

Marina Rinaldi

Isogeometric Analysis (IGA) is a computational methodology for the numerical approximation of Partial Differential Equations (PDEs). IGA is based on the isogeometric concept, for which the same basis functions, usually Non-Uniform Rational B-Splines (NURBS), are used both to represent the geometry and to approximate the unknown solutions of PDEs. Compared to the standard Finite Element method, NURBS-based IGA offers several advantages: ideally a direct interface with CAD tools, exact geometrical representation, simple refinement procedures, and smooth basis functions allowing to easily solve higher order problems, including structural shell problems. In these contexts, repeatedly solving a problem for a large set of geometric parameters might lead to high and eventually prohibitive computational cost. To cope with this problem, we consider in this work the Reduced Basis (RB) method for the solution of parameter dependent PDEs, specifically for which the NURBS representation of the computational domain is parameter dependent. RB refers to a technique that enables a rapid and reliable approximation of parametrized PDEs by constructing low dimensional approximation spaces. In this work, for the construction of the reduced spaces we adopt two different strategies, namely the Proper Orthogonal Decomposition and the greedy algorithm. In this thesis we combine RB and IGA for the efficient solution of parametrized problems for all the possible cases of NURBS geometrical parametrizations, which specifically include the NURBS control points, the weights, and both the control points and weights. In particular, we first focus on the solution of second order PDEs on parametrized lower dimensional manifolds, specifically surfaces in the three dimensional space. We consider geometrical parametrizations that entail a nonaffine dependence of the variational forms on the spatial coordinates and the geometric parameters. Thus, depending on the parametrization at hand and in order to ensure a suitable Offline/Online decomposition between the reduced order model construction and solution, we resort to the Empirical Interpolation Method (EIM) or the Matrix Discrete Empirical Interpolation Method (MDEIM), by comparing their performances. As application, we solve a class of benchmark structural problems modeled by Kirchoff-Love shells for which we consider NURBS geometric parametrizations and we apply the RB method to the solution of this class of fourth order PDEs. We highlight by means of numerical tests, the performance of the RB method applied to standard IGA approximation of parametrized shell geometries.

2015

Differentiable Signed Distance Function Rendering

Wenzel Alban Jakob, Sébastien Nicolas Speierer, Delio Aleardo Vicini

Physically-based differentiable rendering has recently emerged as an attractive new technique for solving inverse problems that recover complete 3D scene representations from images. The inversion of shape parameters is of particular interest but also poses severe challenges: shapes are intertwined with visibility, whose discontinuous nature introduces severe bias in computed derivatives unless costly precautions are taken. Shape representations like triangle meshes suffer from additional difficulties, since the continuous optimization of mesh parameters cannot introduce topological changes. One common solution to these difficulties entails representing shapes using signed distance functions (SDFs) and gradually adapting their zero level set during optimization. Previous differentiable rendering of SDFs did not fully account for visibility gradients and required the use of mask or silhouette supervision, or discretization into a triangle mesh. In this article, we show how to extend the commonly used sphere tracing algorithm so that it additionally outputs a reparameterization that provides the means to compute accurate shape parameter derivatives. At a high level, this resembles techniques for differentiable mesh rendering, though we show that the SDF representation admits a particularly efficient reparameterization that outperforms prior work. Our experiments demonstrate the reconstruction of (synthetic) objects without complex regularization or priors, using only a per-pixel RGB loss.

ASSOC COMPUTING MACHINERY2022

Reduced Basis method for Isogeometric Analysis: application to structural problems

Marina Rinaldi

2015