Global matrices

SOLVER Assembles elemental stiffness equations into a banded global matrix, imposes boundary conditions and solves the set of banded equations using the LU decomposition method (Gerald and Wheatley, 1984). SOLVER calls the following 4 subroutines. ... 

Figure 7.1 Arrangement of the global matrix in the band-solver subroutine...

Each of the five steps demonstrated different parallel efficiencies. Steps two and four were the most intensive in terms of computation. The implementation by Kuppermann and co-workers obtained better than 80% efficiency for up to 64 nodes on the Caltech/JPL Mark Illfp MIMD computer for step two.275 For large global matrix dimensions, step four was 40% efficient on 64 nodes and —80% efficient on 8 nodes. 

From the local matrices pr the "global" matrix PR can be written as follows ... 

The global matrix-equation which takes in the n equations like (6-25), (/ = 1, 2,..., n) is then... 

Now we add up aU the contributions from each element into a global matrix and global vector. 

For a mesh of ne elements (Fig. 6), a global matrix (Fig. 7) is now constructed with contributions from each node summed into the relevant matrix position. 

Pseudo code 1 Generation of the global matrix [GM], boundary condition matrix [5], and global weight matrix [zl]... 

Table 4 also contains the modest memory requirements for storage of the lower triangular portions of each global matrix. In the case of lysozyme, which was the largest structure that could be done on our 32-Mbyte machine, a standard MO calculation with no cutoff would need 98.5 Mbytes of memory to store the lower half of P, H, or F. The use of a cutoff actually linearizes the storage requirements, so essentially all of the bottlenecks are removed for calculations on progressively larger systems. 

The actual assembly is a little more complex (you need to loop over all elements, calculate the integrals above and put the numbers to the correct place in the global matrix), but the above sketch should make the main idea clear. 

The dynamic stiffness matrices and shape functions used in SEM are exact within the scope of the underlying physical theory, and the method allows a reduced number of degrees of freedom. The matrices are depended on frequency, but using spectral analysis, the dynamic response can be easily composed by wave superposition. Harmonic, random, or damped transient excitations can be decomposed using the discrete Fourier transform (DFT). The discrete frequencies are used to calculate the spectral matrix and discrete responses. Then, the complete dynamic response is computed by the sum of frequency components (inverse DFT). As FEM, SEM uses the assembly of a global matrix using elementary matrices and spatial discretization. However, differently from FEM, only discontinuities and locations where loads are applied need to be meshed (Ahmida and Arruda 2001). 

Elementary matrix of straight and curved beams, rod under axial force, and torsion were developed using spectral finite element approach. This method can be used to analyze the stmctural dynamic behavior. From the perspective of earthquake engineering, SEM allows to design structures subject to seismic excitation. The approach has advantages such as low computational cost, acctrracy in frequency/time response, and simple modeling of complex structures. The equations of motion, spectral relation, and the spectral elementary matrix were derived. The simple elements were combined into a 3D frame element and a global matrix was assembled. Time and frequency responses of 3D and 2D frame structures under a... 

A typical finite element simulation proceeds as follows. The domain of the problem is divided into a set of non-overlapping regions termed elements. The solution of the problem is then sought in the approximate form of simple functions such as polynomials over each element. The weak or integral form of the governing equations is used to construct the approximate problem, which is linear if the problem is linear. The contributions from each element are assembled into a global matrix that represents, loosely, the stiffness of the system. 

It is convenient to adopt a local numbering system when evaluating the element stiffness matrix and load vector, in which the nodes are numbered in a counter clockwise direction, starting with node 1. Once the element stiffiiess matrix and load vector have been evaluated, the components can then simply be plaeed in the correct rows and columns of the global matrix and vector by recalling the global node numbers of the element. The process whereby and are computed for each element, and then added to the global matrix, is known as assembly. 

For axisymmetric layers and axial positions of the origins O (along the 2 -axis of rotation), the scattering problem decouples over the azimuthal modes and the transition matrix can be computed separately for each m. Specifically, for each layer /, we compute the Qi matrices and assemble these matrices into the global matrix A. The matrix A is inverted, and the blocks 11 and 21 of the inverse matrix are used for T-matrix calculation. Because A is a sparse matrix, appropriate LU-factorization routines (for sparse systems of equations) can be employed. 

An important feature of this solution method is that the expansion orders of the surface field approximations can be different. To derive the dimension of the global matrix A, we consider an axisymmetric particle. If ATrank(0 is the maximum expansion order of the layer I and, for a given azimuthal mode m, 2Afmax(0 X 2Amax(0 IS the dimension of the corresponding Q matrices, where... 

In practical computer calculations it is convenient to consider the global matrix A with block-matrix components... 

If the number of particles increases, the dimension of the global matrix A becomes excessively large. Wang and Chew [250,251] proposed a recursive T-matrix algorithm, which computes the T matrix of a system of n components by using the transition matrices of the newly added q components and the... 

According to the node superposition principle, the global matrix can be constructed after the element calculation. The velocity solution will be obtained by solving the equations set after the boimdary condition being introduced. Thereafter the velocity value of each... 

---