## [Qj [Ri

Table 5.14 Numerical Recipe's Subroutine HQR c This subroutine is extracted from NUMERICAL RECIPES c (FORTRAN version, 1990, ISBN 0-521-3 8330-7), c Authors Press, Flannery, Teukolsky and Vetterling c Find all eigenvalues of an N by N upper Hessenberg c matrix A , that is stored in an NP by NP array. On c input A can be exactly as output from routine ELMHES, c on output it is destroyed. The real & imaginary parts c of the eigenvalues are returned in WR & WI, respectively, implicit...

## 20 00 10 00 00 110

Line 012 The current i row of U is factorized. Line 014 All previously factorized kth rows will be used to partially factorize the current Ith row. Line 020 The ith row is partially factorized by the previous k1 row. This operation involves a constant ( -xmult) times a vector ( column j, corresponding to row k of U ), and then adds the result to another vector ( column j, corresponding to row i of U ). Lines 25 - 27 Output the upper-triangular portions of the factorized matrix U , according to...

## Info

In other words, the exact first two eigen-vectors (J)'1* and < j> (2) can be expressed as linear combinations of the columns of the starting iteration vectors X, . Thus, subspace iteration algorithm converges in just one iteration. 5.7 Lanczos Eigen-Solution Algorithms For a large-scale system, if one has interest in obtaining the first few eigen-pair solutions for the generalized eigen-equation (5.1), then sub-space iteration algorithms can be employed. However, if the number of requested...

## K K

(1) The prescribed Dirichlet dof, due to the physical supports in subdomains r 1 and 2, are assumed to be already incorporated into Km , m(1) , K(2) , and l (2) in Eqs. (6.317 and 6.318). (2) Compatibility requirements can be imposed on the constrained remainder dof' as follows (see Figure 6.11) (2) Compatibility requirements can be imposed on the constrained remainder dof' as follows (see Figure 6.11) (3) Corner point A is usually selected at a point where more than two sub-domains meet. While...

## Ff Jjcclxdy JJftyjdxdy1264

Again, Eqs.( 1.263 - 1.264) are the same as J. N. Reddy's Eq.(8.14b) 1.1 , C1 -al)'c2 a22'c3 c4 c5 c6 cll c12 0, n C1 fl0,C8 - ,C9 0,C 0 -C y > ) Then, Eq.(1.222) will be simplified to J. N. Reddy's Eq.(8.154) 1.1 . Substituting Eq.( 1.265) into Eqs.( 1.234, 1.235, 1.237), one obtains Kl JJ (ai1 aTlk *22 + (1.266) Cfj - JJc10 i jdxdy JJc j jdxdy (1.267) fi J c8 idxdy JJfyjdxdy (i ,268) Eqs.(1.266 - 1.268) are the same as J. N. Reddy's Eq.(8.159c) l.l , Case 3 Let's define C1 ail'C2 a22'C3 C4...

## 2 1 1

The shifted stiffness matrix j can be computed as The new eigen-problem is defined as Substituting Eq.(5.44) into Eq.(5.45) to get Comparing the new eigen-problem of Eq.(5.46) with the original eigen-problem shown in Eq.(5.1), one has Thus, the eigen-vectors of Eq.(5.1) and Eq.(5.46) are the same, and the eigen-values of Eqs.(5.1, 5.46) are different only by the shifted value p. For the numerical data shown in Eq.(5.43), one has

## 1

The FORTRAN code for this algorithm is given in Table 2.5. _Table 2.5 Matrix Times Vector (Dot Product Operations)_ Do 1 I 1, N (say 4) Do 1 J 1, N Algorithm 2 Dot product operations with unrolling techniques We can group a few (say NUNROL 2) rows of A and operate on a vector x . Thus, algorithm 1 can be modified and is given in Table 2.6. Table 2.6 Matrix Times Vector (Dot Product Operations, with Unrolling Level 2) Do 1 I 1, N, NUNROL ( 2) Do 1 J 1, N b(I)...

## N

Y(x) y(x) Ai )i(x) + (t)o(x) (1.51) i l Based on the discussion in Section 1.3, one obtains from Eq.(1.48) 2n 2 ( the highest order of derivative) (1.52) Thus, Eq.(1.49) and Eq.(1.50) represent the geometrical and natural boundary conditions, respectively. The non-homogeneous geometrical (or essential) boundary conditions can be satisfied by the function < )0(x) such as ( > 0( x0) y0 . The functions < t> j(x> are required to satisfy the homogeneous form of the same boundary condition...

## Stop

C INITIAALIZE SWEEP COUNTER AND BEGIN ITERATION c NSWEEP 0 NR N-1 40 NSWEEP NSWEEP+1 c IF (IFPR.EQ. 1) WRITE (*,2000)NSWEEP c c CHECK IF PRESENT OFF-DIAGONAL ELEMENT IS LARGE ENOUGH TO REQUIRE c ZEROING c EPS (.01 **NSWEEP)**2 DO 210 J 1,NR JJ J+1 EPTOLB (B(J,K)*B(J,K)) (B(J,J)*B(K,K)) IF (( EPTOLA.LT.EPS).AND.(EPTOLB.LT.EPS)) GO TO 210 AKK A(K,K)*B(J,K)-B(K,K)*A(J,K) AJJ A(J,J)*B(J,K)-B(J,J)*A(J,K) AB A(J,J)*B(K,K)-A(K,K)*B(J,J) CHECK (AB*AB+4.*AKK*AJJ) 4. IF (CHECK)50,60,60 50 WRITE(*,2020)...

## Jvc

At x > the exact solution is given by Eq.(l.14) -5 coL* At x , the approximated solution is given by Eq.(l .29) -5oif (a) The selected function < )j (x) can also be easily selected as a polynomial, which also satisfies the geometrical boundary conditions (see Eq. 1.2 ) y(x) (Ai+A2x)*4> ,(x) d-33) y(x) A,( > ,(x) + A24> 2(x) (1.34) < j> 2(x) x< J> , (x) (1.35) (b) If the function < > (x) has to satisfy the following hypothetical boundary conditions, then a possible...

## Gx

Equation(6.142) can be solved iteratively, based on the fact that PF is symmetrical on the space of Lagrange multipliers A, satisfying the constraint G A 0 . (d) The homogeneous form of the constraint GjA 0 (see Eq. 6.142 ) implies that if Aa , and Af) satisfies the homogeneous constraint (meaning GjAa 0 GfAb ), then so does Aa + Ah (meaning (e) The above remark is important, especially in referring to Eq.(6.134). (f) Note that A Range G,), so that, according to Eq.(6.134), we can write

## End

5.7.8 Efficient Software for Lanezos Eigen-Solver A brief description of the highly efficient sparse eigen-solver is given in the following section. User Manual for Sparse Lanezos Eigen-Solver (This file is stored under cee SPARSEPACK97-98 (under file name complex_SPARSEPACKusermanual) 01 To solve the generalized eigen-problem in the form A complex numbers, symmetrical (positive) STIFFNESS matrix B complex numbers, symmetrical MASS matrix Note B can be either diagonal (lumped) mass, or have the...

## 257

From the algorithms presented in Table 5.16, one concludes that major time-consuming operations will be involved with (a) Since Eq.(5.291) involves with A B 1 * C (5.276, repeated) A q(1) Br' * C *q(I) Thus, matrix-vector multiplication for C q(I) is required. Factorization of the unsymmetrical matrix B is required and the forward backward solution is also required. (b) Dot product operations of two vectors is required (see Eq. 5.292 ). (c) Re-orthononalizaton process may be required (see step...

## DM zM PA

Based upon the primal DD formulation, discussed in Sections 6.1 - 6.6, the MPI Fortran software package DIPSS (Direct Iterative Parallel Sparse Solver) has been developed to solve few large scale, practical engineering problems that are summarized in the following paragraphs Example 1 - Three-dimensional acoustic finite element model. In this example, DIPSS is exercised to study the propagation of plane acoustic pressure waves in a 3-D hard wall duct without end reflection and airflow. Figure...

## B

Figure 1.1 a Simply Supported Beam under Uniformly Applied Load If we pretend the right-hand side f in Eq.(1.3) represents the force, and the unknown function y represents the beam deflection, then the virtual work can be computed as Sw JJJ(f)*5y dv JJj(L y)*8y dv (1.6) In Eq.(1.6), 8y represents the virtual displacement, which is consistent with (or satisfied by) the geometric boundary conditions (at the supports at joints A and B of Figure 1.1, for this example). In the Galerkin method, the...

## 2 0 0 1

(discussed in Section 5.2), find all the eigen-pair solution for the generalized eigen-equation shown in Eq.(5.1). 5.2 Using the data for K 4x4 and M 4x4 as shown in Eqs.(5.28 - 5.29), find all the eigen-pair solutions for the generalized eigen-equation by the Generalized Jacobi method. 5.3 Re-do problem 5.2 using the Sub-space Iteration method, with the initial guess for the matrix Xk as

## T

Figure 3.7 Storage Scheme for Unsymmetrical Matrix To illustrate the usage of the adopted storage scheme, let's consider the matrix given in Eq.(3.151). The data in Eq.(3.150) will be represented as follows IA (1 7 neq+ 1) 1,3,4,5,6, 7,7 JA (1 6 ncoef) 4, 6,5,5,5, 6 AD (1 6 neq) 11., 44., 66., 88., 110., 112. AN (1 6 ncoef) 1., 2., 3., 4., 5., 7. AN2(1 6 ncoef) 8., 9., 10., 11., 12., 14. where neq is the size of the original stiffness matrix and ncoef is the number of nonzero, off diagonal...

## 220 00 770

Nreord 3 (use MMD reorder algorithm), 0 (NOT use re-order algorithm) neig number of requested eigen-values vectors (above threshold -shift) lump 1 (lumped, diagonal mass matrix B , 0 (consistent mass matrix B ) n size of matrix A(n,n) same definition as given above n2 n ncoef same definition as given above itime 0 (save memory), 1 (save time) ishift 0 (default, noshift), NONZERO (perform a shift of value ISHIFT) iblock -1 (sub-space iteration algorithm), 0 (regular Lanczos algorithm) mread...

## 160

Panel flutter, 355 Papadrakakis, M., 525 Parallel (MPI) Gram-Schmidt OR, 361 Parallel block factorization, 83 Parallel Choleski factorization, 80 Parallel computer, 64 Parallel dense equation solvers, 77 (D.D.), 396, 397 Preconditioning matrix, 393 Prescribed boundary conditions, 386 Press, Flannery, Teukolsky and Vetterling, 339 Primal DD formulation, 464 Primary dependent function, 32 Primary variable, 9,18 Processor, 64 Projected residual, 421 Proportional damping matrix, 13 Pseudo force,...

## Ncoef2

Table 3.18 HSCT FEM Summary of Results for UNSYNUMFA 1 2 8 Using UnsyMMD and Different Level of Loop Unrolling on the IBM RS6000 590 Stretch Table 3.18 HSCT FEM Summary of Results for UNSYNUMFA 1 2 8 Using UnsyMMD and Different Level of Loop Unrolling on the IBM RS6000 590 Stretch

## 6

NN Size of Element Matrix ( 4, for 2-D truss element, see Figure 4.1) B values of applied nodal loads and or values of prescribed Dirichlet boundary conditions (4.21) 1 The 4x4 element stiffness matrix (for 2-D truss element) should be computed and stored as a 1-D array (column - by - column fashion), such as locate i + (j -1)*(NN 4) i,j l-> 4 2 For the example data shown in Figure 4.1, the vector B should be initialized to

## Y

Thus, Eq.(6.511) represents the implementation of Eq.(6.464) rDl How to Efficiently Handle Successive Right-Hand-Side Vectors'6 23 6 243 Assuming we have to solve for the following problems In Eq.(6.512), the right-hand-side (RHS) vectors may NOT be all available at the same time, but that bj depends on Xj_j . There are two objectives in this section that will be discussed in subsequent paragraphs (1) Assume the 1st solution X , which corresponds to the 1st RHS vector, bj has already been found...

## 25 2 2

Observing the Eq.(5.11), one recognizes that the two equations are NOT independent, and therefore, there is only one ( n-l 2-1) independent equation. Since there are two unknowns ( Q and ) and only one independent equation, there is an infinite number of solutions to Eq.(5.11). A solution can be found by selecting, for example Then the other unknown < > ' can be found from Eq.(5.11) as Similarly, substituting Eq.(5.10) into Eq.(5.6), and selecting Thus, the eigen-matrix < 1> can be...

## 1 0 0 1 0 0 0

6 Finite Element Domain Decomposition Procedures The finite element (statics) equilibrium equations can be given as b design variable vector such as cross-sectional areas of truss members, moment of inertia of beam members, and or thickness of plate (or shell) members. Z nodal displacement vector K stiffness matrix where subscripts B and I denote Boundary and Interior terms, respectively. From the 2nd part of Eq.(6.2), one has From the 1st part of Eq.(6.2), one has Substituting Eq.(6.4) into...

## A

Figure 1.3 Local-Global Coordinate Transformation Figure 1.3 Local-Global Coordinate Transformation x A cos(9) + yA sin(0) y A cos(8) - x A sin(9) j Substituting Eqs.(l .74, 176) into Eq.(l .71), one has k' * A, r M p Pre-multiplying both sides of Eq.(1.77) by X T, one gets Since I , thus X T , hence Eq.(1.78) becomes k X, T k' X k(e) element stiffness in global coordinate reference (1.81) The finite element stiffness equations, given by Eq.(1.80), can be assembled to form the following system...

## 0032

4.4 Symbolic Sparse Assembly of Symmetrical Matrices Assuming the element-dof connectivity matrix E (see Eq. 4.6 ), the dof-element connectivity matrix E T (see Eq. 4.8 ), and the locations of the Dirichlet boundary conditions vector ibdof (see Eq. 4.7 ) are known. The non-zero patterns of matrices E and E T can be described by the integer arrays ie , je and iet , jet as indicated in Eqs.(4.4,4.5), and Eqs.(4.9,4.10), respectively. To facilitate subsequent discussion, information about the...

## 4 6 4 6 9 6 4 6

Once sub-matrices and 2 J can be identified, the generalized inverse matrix, and the corresponding rigid body matrix can be computed from Eq.(6.110), and Eq.(6.112), respectively. Both Eqs.(6.110, 6.112) require the computation of the factorized . For efficient computation of the factorized < fn and recovering the original sub-matrix CI2 , the following step-by-step procedure is recommendedI6U1. Step 1 The floating sub-domain's stiffness matrix J (see...

## F F

Using the new notations, Eqs.(6.365, 6.367) can be expressed as (in terms of the unknowns and uc) The unknown vector mc can be eliminated from the 2nd half of Eq.(6.371) as follows Substituting Eq.(6.372) into the 1st half of Eq.(6.371) (5) Matrix Kec is sparse, symmetric positive definite, and the size of this matrix is relatively small. (6) Matrix A is non-singular since there will be enough selected corner dof' to guarantee that Krr exists. The entire FETI-DP step-by-step procedures can be...

## Q

Eq.(1.201) can be integrated by parts once, to give Then, Eq.(1.203) can be re-written as dcodw dcodW a (Odw q +c2 +c3 ---Wf The primary dependent function a can be assumed as 0) Yj )jNej(x, y,z) N(x,y,z)lm i i In Eq.( 1.206), n, (0j, and Nj represent the number of dof per element, element nodal displacements, and element shape functions, respectively. For a 4-node tetrahedral element (see Figure 1.9) n 4, the assumed field can be given as co(x, y, z) aj + (a2x + a3y + a4z) For an 8-node brick...

## Nel

P p(e) and p(e) is the right-hand-side of Eq.(l.80) R system unknown displacement vector For structural engineering applications, the system matrix K is usually a sparse, symmetrical, and positive definite matrix (after imposing appropriated boundary conditions). At this stage, however K is still a singular matrix. After imposing the appropriated boundary conditions, K will be non-singular, and the unknown vector R in Eq.(1.82) can be solved. 1.6 Flowcharts for Dynamics Finite Element Analysis...

## 0 0 0 0 0 0 1 0 0

Following exactly the same derivations given in the previous section, the generalized version of Eqs.(3.112) and (3.118) can be given as (a) If a is a symmetrical matrix (which it is ), then a ' is also symmetrical. One starts with the following identity AM W XrtA (3.126) Transposing both sides of Eq.(3.126), one obtains A-'fMMl (3-127) Since both matrices a and i are symmetrical, Eq.(3.127) becomes Post-multiplying both sides of Eq.(3.129) by a-1 J, one has a-'MA1 (3.130) Thus, a-1 is also a...

## 12

It should be noted that sizes of arrays icolst and (icolendj are the same for the same processor, but may be different from other processors. In fact, it depends on how many blocks are assigned to each processor (noblk). For this example, processors 1 and 2 each stores 3 blocks of the matrix, and processor 3 stores 2 blocks of the matrix. The stiffness matrix, stored in a one-dimensional array A and the right-hand-side load vector b will be automatically generated such that the solution vector...

## Due Thai Nguyen

Old Dominion University Norfolk, Virginia Prof. Due Thai Nguyen 135 Kaufman Old Dominion University Department of Civil & Environmental Engineering Multidisc. Parallel-Vector Comp Ctr Norfolk VA 23529 Finite Element Methods Parallel-Sparse Statics and Eigen-Solutions Library of Congress Control Number 2005937075 ISBN 0-387-29330-2 e-ISBN 0-387-30851 -2 ISBN 978-0-387-29330-1 2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or...

## So

A< k> '2 a< k> - q Ta(k) qW - q 2> V> q(2> Eqs.(5.255 - 5.256) in the modified Gram-Schmidt are, therefore, equivalent to the classical Gram-Schmidt ( to project the original vector a ' onto both q(1) and q< 2)). In the classical Gram-Schmidt, the current vector q(k) is orthogonalized with all previous vectors q(1), q< 2), , q< k l). However, in the modified Gram-Schmidt, the current vector, say q'1is orthogonalized with subsequent vectors a00'1 q< 2), a(k)'2 q(3), etc....

## 211110

(c) Kll.INDXS file, to store column numbers associated with the non-zero, off-diagonal terms (in a row-by-row fashion) of the STIFFNESS A matrix (integer numbers, should have NCOEF values) (d) K.DIAG file, to store numerical values of DIAGONAL terms of matrix A (could be REAL or COMPLEX numbers, should have N values) (11.0,0.0), (44.0,0.0), (66.0,0.0), (88.0,0.0), (100.0,0.0), (12.0,0.0) (e) Kll.COEFS file, to store numerical values of the non-zero, offdiagonal terms (in a row-by-row fashion)...

## Ok

As the next step, the user is supposed to compile his her program and create the executable file. To do this From the pulldown menu, go to Build, then select Compile to compile the MPI code. When you do this, a folder named Debug will be created. Go to Build again, and select Build to create the executable of the MPI code. The executable file will be located under the folder Debug. H. Specifying the Computer Name(s) In order to specify the machine name, create a file named Machines under the...

## Ja

Fc After exiting subroutine symbass, we will obtain the final output arrays iaj and ja as shown in Eq.(4.11), and Eq.(4.12), respectively. The number of non-zero, off-diagonal terms of the upper-triangular matrix K , shown in Eq.(4.3), can be computed as NCOEF1 IA (N+l)-1 7-1 6 (4.17) 4.5 Numerical Sparse Assembly of Symmetrical Matrices To facilitate the discussion in this section, the following variables and arrays are defined (refer to the example, shown in Figure 4.1). Input IA, JA (see...

## Dir

Figure 1.11 a 2-D Rectangular Element with Its Local Coordinates ve (i_i)(i_Z), e2 x(i-i), V4 a--)r o-243) To facilitate the computation of Eq.(1.234), K can be expressed as the sum of several basic matrices Sa (a, 0,1,2). where denotes the transpose of the enclosed matrix or vector, and f JVi,< x j, dxdy (1.247) i,a 'xi x'x2 y Vi,o i> j,o Vj (1-248) Using the Linear Triangular Element and let

## 8595

Example 2 - Three-dimensional structural bracket Unite element model. The DD formulation has also been applied to solve the 3-D structural bracket problem shown in Figure 6.7. The finite element model contains 194,925 degrees of freedom (N 194,925) and the elements in the matrix, K, are real numbers. Results were computed on a cluster of 1 - 6 personal computers (PCs) running under Windows environments with Intel Pentium. It should be noted that the DIPSS software was not ported to the PC...

## Ulo

For stable substructures, K is just the regular inverse of the K(s' matrix. In this step, we will factorize Kn of each substructure. ODU's symmetrical direct sparse solver, MA28, MA47 or SuperLU 1.9,4.2,3.2,6.8 can be used for this purpose. 6. Find the rigid body modes R of each substructure. In this step, there will be no R for stable substructures.

## AiA1

Ai e Kernel (Gj) and A1 e Range Gj) Pre-multiplying the top portion of Eq.(6.118) by P, one gets Utilizing Eq.(6.133), the above equation will reduce to Thus, the dual interface Eq.(6.118) is transformed to The key idea of the CPG is to select the proper initial value A0 so that it will automatically guarantee to satisfy the (constrained) condition (shown in the 2nd half of Eq.6.138 ) in subsequent iterations. A particular solution for A0 is of the form A0 GI(GjGiyi e (6.139) (a) If the...

## 2 0 0

In Eq.(1.106), the superscripts of y represent the Is' eigen-vector (associated with the Is' eigen-value lj) while the subscripts of y represent the components of the 1st eigen-vector. There are two unknowns in Eq.(1.106), namely and y however, there is only one linearly independent equation in (1.106). Hence, if we let The 1st eigen-vector < ) (associated with the 1st eigen-value therefore, is given as Similar procedures can be used to obtain the 2na eigen-vector (j)'2-1 (associated with the...

## Kmr

The kinetic energy can also be expressed as Comparing Eq.(1.85) with Eq.(1.86), the local element mass matrix m' can be identified as m m JfNixOftNixOJpdV fm' (1.87) The global element mass matrix m can be obtained in a similar fashion as indicated in Eq.(1.81), hence m WT m'(e) X m(e) (1.88) The system mass matrix can be assembled as NEL where m(e> has already been defined in Eq.(1.88). The system dynamical equilibrium equations, therefore, can be given as M R + K R P(t) (1.90) If a...

## P4

Eq.(1.178) is more preferable as compared to Eq.(1177), due to the following reasons (i) The modified system stiffness matrix K is non-singular (ii) The modified right-hand-side vector P is completely known Thus, the unknown displacement vector efficient algorithms and software, which are available in different computer platforms. Detailed descriptions of sparse equation solution algorithms 1.9 will be discussed in Chapter 3. Having obtained all the unknown global displacement vector solving...

## Neltypes

Ncoefl ie(8)-1 33-1 32 NDOFPE *NELTYPE(i) where NLETYPE(l), and NELTYPE(2) represents the number of finite element type 1 ( 2-D truss element), and type 2 ( 2-D triangular element), respectively. The general flowchart for symmetrical sparse assembly process for mixed finite element model is outlined in Figure 4.4. call sparseasseml (please see the attached subroutine in Table 4.6) Figure 4.4 Outlines for Symmetrical Sparse Assembly Process with Mixed Finite Element Types Using the computer...

## R

We need sparse matrix times matrix subroutine for this operation. The size of the resulted matrix will be (ninfeq, ndofs-rank) F, Y,B(S)K(S) B(s) Hence, F, will be a square matrix whose size is (ninfeq, ninfeq). Matrix times matrix subroutine will be required for the above operations. G, fl(,) (,) z*< 2, (2) J flW) W) - Thus, G, will be a matrix whose size is (ninfeq, summation of (ndofs-rank) for all substructures). . Thus, the size of d will be the number of interface dofs

## 784

It should be mentioned that fill-in terms associated with Eq.(4.70) are NOT shown here. Kmetis,be * metis, be i -metis, be) 02,0.975,0.996, 0,5,1.575,1.265,0.656,1.490,04 (4.73) which have been independently confirmed by the developed computer program shown in Table 4.9. Table 4.9 Unsymmetrical Sparse Assembly c Table 4.9 Unsymmetrical Sparse Assembly c subroutine parti unsym implicit real*8(a-h,o-z) c purpose d.t. nguyen's general, unsymmetrical sparse assembly c author Prof. Due T. Nguyen...

## Civil Engineering Next Processor

2.6.9 Block Error Checking Subroutine After the solution of the system of equations has been obtained, the next step that should be considered is error checking. The purpose of this phase is to evaluate how good the obtained solution is. There are four components that need to be considered 1. X is the X that has the maximum absolute value (i.e., the maximum displacement, in structural engineering application). 3. Absolute error norm of A X b 4. Relative error norm is the ratio of absolute error...

## [aMbJMc

And the original solution can be obtained as Since only a few of the lowest eigen-pairs ( 2,0 ) are required, a reduction scheme that makes the largest ones ( (i, x ) converge first is preferred. This can be done by defining the orthogonal transformation matrix Qm ( 1 < m < n ) as QmQm I , Identity matrix (5.280) where Qm qh q2, , qm , q 1, qi is an arbitrary starting vector, and qi+1is determined through hi+l,i1i+l Aq - hk qk (5.281) (hki q ATqk) (i l,2, ,m)j or hk i Aq(i) Tq(k) J...

## 16

- (Area of segment 4) + (Area of segment of 8) (Area)p3 (h * Average height of segment 4) + (h * Average height of segment 8) (2.4) h * (Average height of segment 4 + Average height of segment 8) (2. 5) The values of sum and mypi are computed inside and outside loop 20, respectively (see Table 2.1). Since there are many comments in Table 2.1, only a few remarks are given in the following paragraphs to further clarify the MPI FORTRAN code Remark 1. In a parallel computer environment, and...

## 0504 0574 0644 0761 0057 0646 0408 0816 0408

Example 4 Relationship between SVD and generalized inverse Let the m x n matrix A of rank k have the SVD A UZVH with Oj gt o2 gt -- gt ak gt 0. Then the generalized inverse A of A is the nxm matrix. Ejj lt 5 1 for l lt i lt k and E is the kxk diagonal matrix, with 1.1 J. N. Reddy, An Introduction to the Finite Element Method, 2nd edition, McGraw-Hill 1993 1.2 K. J. Bathe, Finite Element Procedures, Prentice Hall 1996 1.3 K. H. Huebner, The Finite Element Method for Engineers, John Wiley amp...

## 987654321

D. Nguyen and Don N. Nguyen 1. A Review of Basic Finite Element 1.2 Numerical Techniques for Solving Ordinary Differential Equations 1.3 Identifying the Geometric versus Natural Boundary 1.4 The Weak 1.5 Flowcharts for Statics Finite Element 1.6 Flowcharts for Dynamics Finite Element 1.7 Uncoupling the Dynamical Equilibrium 1.8 One-Dimensional Rod Finite Element 1.8.1 One-Dimensional Rod Element Stiffness 1.8.2 Distributed Loads and Equivalent Joint 1.8.3...