Ordinary differential equations finite element method

Other methods can be used in space, such as the finite element method, the orthogonal collocation method, or the method of orthogonal collocation on finite elements. One simply combines the methods for ordinary differential equations (see Ordinary Differential Equations—Boundary Value Problems ) with the methods for initial-value problems (see Numerical Solution of Ordinary Differential Equations as Initial Value Problems ). Fast Fourier transforms can also be used on regular grids (see Fast Fourier Transform ). 

Continuum wavefunctions can also be generated by solving the partial differential equation (3.3) directly without first transforming it into a set of ordinary differential equations. One possible scheme is the finite elements method (Askar and Rabitz 1984 Jaquet 1987). Another method, which has been applied for the calculation of multi-dimensional scattering wavefunctions, is the 5-matrix version of the Kohn variational principle (Zhang and Miller 1990). 

Finite element methods — The finite element method is a powerful and flexible numerical technique for the approximate solution of (both ordinary and partial) differential equations involving replacing the continuous problem with unknown solution by a system of algebraic equations. The method was first introduced by Richard Courant in 1943 [i], and over the next three decades, and particularly in the 1960s, a comprehensive mathematical framework was developed to underpin the method. 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

The GRM using the generalized Maxwell-Stefan equations has no closed-form solutions. Numerical solutions were calculated using a computer program based on an implementation of the method of orthogonal collocation on finite elements [29,62,63]. The set of discretized ordinary differential equations was solved with the Adams-Moulton method, implemented in the VODE procedure [64]. The relative and absolute errors of the numerical calculations were 1 x 10 and 1 x 10 , respectively. 

Only numerical solutions of the VERSE model can be obtained [65]. The partial differential equations are discretized by application of the method of orthogonal collocation on fixed finite elements. Equation 16.59 is divided into 50 or 60 elements, each with four interior collocation points. Legendre polynomials are used for each element. For Eq. 16.62, only one element is required. It is described by a Jacobi polynomial with two interior collocation points. The resulting set of ordinary differential equations, with their initial and boundary conditions and the chemical equations, are solved using a differential algebraic system solver (DASSL) [65,66]. 

One method to solve partial differential equations using the numerical schemes developed for solving time dependent ordinary differential methods is the method of lines. In this method, the spatial derivatives at time t are replaced by discrete approximations such as finite differences or finite element methods such as collocation or Galerkin. The reason for this approach is the advanced stage of development of schemes to solve ordinary differential equations. The resulting numerical schemes are frequently similar to those developed directly for partial differential equations. 

By using the discrete variable method the 3-dimensional Schrodinger equation is mapped onto a system of ordinary differential equations in the radial coordinate. We solve this system by using the finite element method. 

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

Based on the above assumptions, the model equations are shown in Table 4. The mass balance equations at the pellet and crystal level are based in the double linear driving model equations or bidisperse model[30]. The solution of the set of parabolic partial differential equations showed in Table 4 was performed using the method of lines. The spatial coordinate was discretized using the method of orthogonal collocation in finite elements. For each element 2 internal collocation points were used and the basis polynomial were calculated using the shifted Jacobi polynomials with weighting function W x) = (a = Q,p=G) hat has equidistant roots inside each element [31]. The set of discretized ordinary differential equations are then solved with DASPK solver [32] which is based on backward differentiation formulas. 

For either numerical solution of the field equations by means of the finite element method or determination of a system of ordinary differential equations for modal amplitudes, the existence of a variational statement or weak form of the field equations is essential. For the complementary aspect of the problem concerned with the elastic field for a fixed boundary configuration, the powerful minimum potential energy theorem is available (Fung 1965). The purpose here is to introduce a variational principle as a basis for describing the rate of shape evolution for a fixed shape and a fixed elastic field. 

An equivalent formulation of finite element methods can be developed using the concept of weighted residuals. In Sec. 5.6.3, we discussed the method of weighted residuals in connection with the solution of the two-point boundary-value problem. In that case we chose the solution of the ordinary differential equation as a polynomial basis function and caused the integral of weighted residuals to vanish ... 

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