Lecture

Finite element methods

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Description

This lecture introduces finite element methods for solving diffusion problems in porous media. It covers topics such as meshing, interpolation, method of weighted residuals, steady-state diffusion equations, boundary conditions, and the implementation of a finite element program. The instructor explains the transformation of continuum problems into a finite number of equations using a mesh, the concept of trial functions and shape functions for interpolation, and the application of weighted residuals to approximate solutions. The lecture also discusses the assembly of global matrices, solving systems of linear equations, and common pitfalls in space discretization and time integration.

In MOOC

Sorption and transport in cementitious materials

In the MOOC: “Cement chemistry and sustainable cementitious materials”; you learned how to create more sustainable cementitious materials. But concrete is not an inert rock. It is a dynamic porous material, which can exchange water, gas and ions with its environment. Through physical and chemical interactions, these exchanges can have dramatic consequences: cracks due to shrinkage, reinforcement bars corrosion, spalling, … For a true sustainability, the durability of cementitious materials should not be f

Official source

Ontological neighbourhood

Mathematics

Analysis: Numerical analysis

Related concepts (32)

Interpolation

In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling or experimentation, which represent the values of a function for a limited number of values of the independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable.

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues.

Multivariate interpolation

In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable (multivariate functions); when the variates are spatial coordinates, it is also known as spatial interpolation. The function to be interpolated is known at given points and the interpolation problem consists of yielding values at arbitrary points . Multivariate interpolation is particularly important in geostatistics, where it is used to create a digital elevation model from a set of points on the Earth's surface (for example, spot heights in a topographic survey or depths in a hydrographic survey).

Polynomial interpolation

In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset. Given a set of n + 1 data points , with no two the same, a polynomial function is said to interpolate the data if for each . There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts.

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