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Lecture

Isoparametric Formulation of Quad4 Element

In course

CIVIL-321: Numerical modelling of solids and structures

La modélisation numérique des solides est abordée à travers la méthode des éléments finis. Les aspects purement analytiques sont d'abord présentés, puis les moyens d'interpolation, d'intégration et de résolution de la mécanique sont étudiés. ...

Description

This lecture covers the isoparametric formulation of the Quad4 planar element, explaining the process of calculating stiffness matrix based on the reference element (s, t) rather than specific element cases (x, y). It includes choosing the element, interpolation functions, deriving the stiffness matrix, and finding consistent forces. The lecture also discusses the interpolation functions, matrix Jacobian, gradients calculation, and assembly algorithm for stiffness matrix. Additionally, it explores the physical interpretation of the Jacobian determinant, the algorithm for consistent forces calculation, and the general algorithm for stiffness matrix assembly. The lecture concludes with the explanation of the forces in natural coordinates and the challenges of manual integration in numerical methods.

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Instructors (2)

Guillaume Anciaux

Jean-François Molinari

Professor J.F. Molinari is the director of the Computational Solid Mechanics Laboratory (http://lsms.epfl.ch) at EPFL, Switzerland. He holds an appointment in the Civil Engineering institute, which he directed from 2013 to 2017, and a joint appointment in the Materials Science institute. He started his tenure at EPFL in 2007, and was promoted to Full Professor in 2012. He is currently an elected member of the Research Council of the Swiss National Science Foundation in Division 2 (Mathematics, Natural and Engineering Sciences), and co editor in chief of the journal Mechanics of Materials. J.F. Molinari graduated from Caltech, USA, in 2001, with a M.S. and Ph.D. in Aeronautics. He held professorships in several countries besides Switzerland, including the United States with a position in Mechanical Engineering at the Johns Hopkins University (2000-2006), and France at Ecole Normale Supérieure Cachan in Mechanics (2005-2007), as well as a Teaching Associate position at the Ecole Polytechnique de Paris (2006-2009). The work conducted by Prof. Molinari and his collaborators takes place at the frontier between traditional disciplines and covers several length scales from atomistic to macroscopic scales. Over the years, Professor Molinari and his group have been developing novel multiscale approaches for a seamless coupling across scales. The activities of the laboratory span the domains of damage mechanics of materials and structures, nano- and microstructural mechanical properties, and tribology.

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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.

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.

Bilinear interpolation

In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., x and y) using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be generalized to functions defined on the vertices of (a mesh of) arbitrary convex quadrilaterals. Bilinear interpolation is performed using linear interpolation first in one direction, and then again in another direction.

Trigonometric interpolation

In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods. This form is especially suited for interpolation of periodic functions. An important special case is when the given data points are equally spaced, in which case the solution is given by the discrete Fourier transform.

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