Development and Application of the Finite Element Method based on MatLab

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Finite Elements in Solid Mechanics. Applications to Convective Transport. Introduction to Fluid Flow. Introduction to Boundary Elements. Introduction to Meshless Methods. It covers a range of finite element applications that nonstructural engineers and scientists from different fields will appreciate either to start learning about them or as a reference for the day-to-day practice. Overall, I think that this is a great book if the reader is looking for a nonstructural approach, that is also easy to follow and that covers a range of topics that can be useful in different engineering fields and for multiphysicalapplications.

Part of the book focuses in the development of the discrete set of algebraic equations in more than one dimensions originated from concrete real-life problems. There are many books in this area. I consider, the third edition of this book to be one of the best for students, researchers and practitioners in this area. You will be prompted to fill out a registration form which will be verified by one of our sales reps. We provide complimentary e-inspection copies of primary textbooks to instructors considering our books for course adoption.

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Finite element method - Wikipedia

We provide a free online form to document your learning and a certificate for your records. Already read this title? Please accept our apologies for any inconvenience this may cause. Nevertheless,theFEM today is a question of economy. On the one hand its industrial application is forced to reduce product development costs and time, on the other hand a large number of commercial FEM codes and a still growing number of software for e?

Due to that, today it is a quite challenging task to operate with all these di?

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So, we want to help in getting a deeper insight into the main "interfaces" between the "customers of the FEM" and the codes itself by providing a totally open structured FE-code based on Matlab, which is a very powerful tool in operating with matrix based formulations. That idea and conditions forced us some years ago to initiateDAEdalon as a tool for general FE developments in research appli- tions.

In spite of still existing high sophisticated - mostly commercial - FE codes, the success and the acceptance of such a structured tool justify that decision afterwards more and more. Read more Read less.

Applicable only on ATM card, debit card or credit card orders. Cashback will be credited as Amazon Pay balance within 10 days. Valid only on your first 2 online payments. Cashback will be credited as Amazon Pay balance within 10 days from purchase. There are various numerical solution algorithms that can be classified into two broad categories; direct and iterative solvers. These algorithms are designed to exploit the sparsity of matrices that depend on the choices of variational formulation and discretization strategy.

Postprocessing procedures are designed for the extraction of the data of interest from a finite element solution. In order to meet the requirements of solution verification, postprocessors need to provide for a posteriori error estimation in terms of the quantities of interest. When the errors of approximation are larger than what is considered acceptable then the discretization has to be changed either by an automated adaptive process or by action of the analyst.

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There are some very efficient postprocessors that provide for the realization of superconvergence. We will demonstrate the finite element method using two sample problems from which the general method can be extrapolated. It is assumed that the reader is familiar with calculus and linear algebra. P2 is a two-dimensional problem Dirichlet problem. The problem P1 can be solved directly by computing antiderivatives. For this reason, we will develop the finite element method for P1 and outline its generalization to P2. Our explanation will proceed in two steps, which mirror two essential steps one must take to solve a boundary value problem BVP using the FEM.

After this second step, we have concrete formulae for a large but finite-dimensional linear problem whose solution will approximately solve the original BVP. This finite-dimensional problem is then implemented on a computer.

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The first step is to convert P1 and P2 into their equivalent weak formulations. Existence and uniqueness of the solution can also be shown. P1 and P2 are ready to be discretized which leads to a common sub-problem 3. The basic idea is to replace the infinite-dimensional linear problem:. One hopes that as the underlying triangular mesh becomes finer and finer, the solution of the discrete problem 3 will in some sense converge to the solution of the original boundary value problem P2.

This parameter will be related to the size of the largest or average triangle in the triangulation. Since we do not perform such an analysis, we will not use this notation.

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Depending on the author, the word "element" in "finite element method" refers either to the triangles in the domain, the piecewise linear basis function, or both. So for instance, an author interested in curved domains might replace the triangles with curved primitives, and so might describe the elements as being curvilinear. On the other hand, some authors replace "piecewise linear" by "piecewise quadratic" or even "piecewise polynomial". The author might then say "higher order element" instead of "higher degree polynomial". Finite element method is not restricted to triangles or tetrahedra in 3-d, or higher order simplexes in multidimensional spaces , but can be defined on quadrilateral subdomains hexahedra, prisms, or pyramids in 3-d, and so on.

Higher order shapes curvilinear elements can be defined with polynomial and even non-polynomial shapes e. More advanced implementations adaptive finite element methods utilize a method to assess the quality of the results based on error estimation theory and modify the mesh during the solution aiming to achieve approximate solution within some bounds from the exact solution of the continuum problem. Mesh adaptivity may utilize various techniques, the most popular are:. Such matrices are known as sparse matrices , and there are efficient solvers for such problems much more efficient than actually inverting the matrix.

For problems that are not too large, sparse LU decompositions and Cholesky decompositions still work well. For instance, MATLAB 's backslash operator which uses sparse LU, sparse Cholesky, and other factorization methods can be sufficient for meshes with a hundred thousand vertices.

A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems , piecewise polynomial basis function that are merely continuous suffice i. For higher order partial differential equations, one must use smoother basis functions. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method , an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint.

Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an h -method h is customarily the diameter of the largest element in the mesh.

If instead of making h smaller, one increases the degree of the polynomials used in the basis function, one has a p -method. If one combines these two refinement types, one obtains an hp -method hp-FEM.

In the hp-FEM, the polynomial degrees can vary from element to element. High order methods with large uniform p are called spectral finite element methods SFEM.

These are not to be confused with spectral methods. The generalized finite element method GFEM uses local spaces consisting of functions, not necessarily polynomials, that reflect the available information on the unknown solution and thus ensure good local approximation. The effectiveness of GFEM has been shown when applied to problems with domains having complicated boundaries, problems with micro-scales, and problems with boundary layers.

Baaser Herbert. Development and Application of the Finite Element Method based on MATLAB

The mixed finite element method is a type of finite element method in which extra independent variables are introduced as nodal variables during the discretization of a partial differential equation problem. The hp-FEM combines adaptively, elements with variable size h and polynomial degree p in order to achieve exceptionally fast, exponential convergence rates. The hpk-FEM combines adaptively, elements with variable size h , polynomial degree of the local approximations p and global differentiability of the local approximations k-1 in order to achieve best convergence rates.

It extends the classical finite element method by enriching the solution space for solutions to differential equations with discontinuous functions. Extended finite element methods enrich the approximation space so that it is able to naturally reproduce the challenging feature associated with the problem of interest: It was shown that for some problems, such an embedding of the problem's feature into the approximation space can significantly improve convergence rates and accuracy.

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Moreover, treating problems with discontinuities with XFEMs suppresses the need to mesh and remesh the discontinuity surfaces, thus alleviating the computational costs and projection errors associated with conventional finite element methods, at the cost of restricting the discontinuities to mesh edges. Several research codes implement this technique to various degrees: It is a semi-analytical fundamental-solutionless method which combines the advantages of both the finite element formulations and procedures, and the boundary element discretization. However, unlike the boundary element method, no fundamental differential solution is required.

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