Manual Programming the Finite Element Method

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Contents:
  1. Характеристики
  2. Programming the Finite Element Method, 5th Edition
  3. Catalog Record: Programming the finite element method : with | HathiTrust Digital Library

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Overview of Finite Element Method (FEM)

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Community Reviews. Showing Rating details. All Languages. More filters. Sort order. Start your review of Programming the Finite Element Method. Nana Danso rated it liked it Sep 16, Souvik Roy rated it it was amazing Jun 03, Sara Elizabeth rated it really liked it Jul 29, Ajit rated it really liked it May 31, DMow rated it really liked it Feb 23, Celal rated it really liked it Oct 09, John rated it really liked it Oct 19, BookDB marked it as to-read Oct 23, Wilco marked it as to-read Feb 01, Dorival marked it as to-read Feb 14, Badr Alshammary marked it as to-read Sep 27, Raj marked it as to-read May 14, Davood marked it as to-read Jan 27, Sukhpreet Singh marked it as to-read Oct 01, Northern added it Apr 03, Trung is currently reading it Jan 23, Dat Le marked it as to-read Apr 28, Very similar to FindRoot , NDSolve needs an initial seeding for each dependent variable when solving stationary nonlinear partial differential equations with the finite element method.

The initial seeding can be specified with InitialSeeding. If an initial seeding is not specified, it is taken to be zero. In the sections that follow, the general internal solution process that NDSolve takes for nonlinear equations like the preceding are presented. The same stages as in the linear case are present in the nonlinear case. The initialization stage for nonlinear equations is mostly equivalent to the linear case. The only difference in the nonlinear case is that, like for FindRoot , a starting seed needs to be specified. This is done by setting the solution data component "DependentVariables" to the initial seed.

The initial seed can be a numeric value or any expression based on the independent variables, and in this case. For highly nonlinear problems, one approach is to solve a less nonlinear problem and use the resulting InterpolatingFunction as a initial seed for the highly nonlinear problem. If no initial seed is specified, is taken as the initial seed.

Характеристики

Solving nonlinear equations is an iterative process. At the lowest level, the iteration process is done by FindRoot.


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The first step is to linearize the PDE coefficients. The details of this process follow. The initial seed is then used to evaluate the linearized PDE. This creates a linear system of equations that can be solved with LinearSolve. The result of that process is fed back into the PDE, and the updated linearized PDE is discretized with the new approximation for the solution.

Programming the Finite Element Method, 5th Edition

This process is repeated until, ideally, the solution converges. The function PDESolve orchestrates the call of FindRoot , takes care of the linearization and the repeated discretization and returns a solution data object that contains the solution or, if the algorithm failed to converge, the best approximation to the solution PDESolve could find. The solution that is found is stored in the solution data object returned.

An InterpolatingFunction object can be created from the solution data.

Catalog Record: Programming the finite element method : with | HathiTrust Digital Library

FindRoot by default uses the Newton — Raphson method to solve nonlinear equations. The way the method works is that it starts with an assumed solution, the initial seed, and tries to improve on it by using the linearized form of the equations. To understand the linearization process better, it is helpful to review the derivation of the Newton method for a scalar function. Let be a nonlinear function of and the unknown solution such that. Now take an arbitrary seed and subtract that from the unknown solution :.

After rearranging the equation to. Taylor expand around and obtain:. Assuming that the derivative of is not zero, you can get to an iteration process:. Generalizing this for an iteration count , you obtain:. This procedure is continued until a solution is found.


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As a next step, the iterative procedure is wrapped in a function that computes a step given a vector. While the norm of two consecutive steps is above a threshold, a new next step is computed. One thing to note in the given example is that both and the Jacobian have constant parts and parts in them that depend on. When solving a nonlinear PDE with the finite element method, it makes sense to compute constant parts once prior to the iterative process and to add them to the nonconstant parts that are computed in each step.

The linearization process for the nonlinear PDE in the coefficient form given in equation 2 follows the same steps. Start from equation 3 and set and to obtain a time-independent coefficient form:. To perform the linearization, now proceed in the same manner as for the scalar case above. Expand with a Taylor series around and obtain:. For and , the derivatives with respect to and are computed:.

The actual solution of the linearized PDE makes use of the coefficient form given in equation 4 with the following coefficients for an arbitrary number of dependent variables:. For scalar equations in ,. After the explanation of Newton's root-finding method and how it applies to partial differential equations, it is instructive to continue to work through the example of solving. In a first step, the linearized coefficients are constructed. The partial derivative coefficients for in equation 7 can be computed by using D.

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The partial derivative coefficients for in equation 8 can be computed by using D. The partial derivative coefficients for in equation 9 can be computed by using D. The partial derivative coefficients for in equation 10 can be computed by using D. These coefficients are conveniently computed with a utility function LinearizePDECoefficients from the finite element context. Once the PDE is linearized, the coefficients are split. This is done because the coefficients are needed in different parts of the root-finding algorithm. The coefficient form. All terms on the left will be assembled into the stiffness matrix.

The stiffness matrix is the Jacobian for the PDE. The stiffness matrix is analog to the term from the preceding scalar example. Since is a tangent to , the stiffness matrix is also called the tangent stiffness matrix.

The terms on the right of the PDE correspond to. The function femRHS given here also takes care of generalized Neumann values even though none are present in the example under consideration. In fact, the function given here is very close to the actual implementation. Note that in function femRHS only the load vector is made use of. Once the variables nonlinear and nonlinearLoad are no longer needed, they are set to Null to free the memory they point to.

Again, the function femJacobian is very close to the actual implementation and variables no longer needed are set to Null. For the construction of the Jacobian, the linearized stiffness components are used. These include the diffusion coefficient , the conservative convection coefficient , the convection coefficient and the reaction coefficient. It is noteworthy that in the implementation, all coefficients that are linear will only be assembled once. In both the functions femRHS and femJacobian , only the nonlinear coefficients will be computed and added to the previously computed linear components.

This avoids unnecessary computations of linear coefficients that remain unchanged in each iteration. Last, the nonlinear and linear boundary conditions are deployed. The last step is to find the solution to the linearized PDE. For this an affine covariant Newton method is used.