Hutton - Fundamentals of Finite Element Analysis (523155), страница 86

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For large systems of equations, calculation of the inverse of the coefficient matrix is time consuming and expensive. Fortunately, the operation ofinverting the matrix is not necessary to obtain solutions. Many other methods aremore computationally efficient. The method of Gauss elimination is one suchtechnique. Gauss elimination utilizes simple algebraic operations (multiplication,division, addition, and subtraction) to successively eliminate unknowns from asystem of equations generally described by   a11 a12 · · · a1n   f1  x1  a21 a22 · · · a2n  x2   f2 =[A]{x} = { f } ⇒  ..(C.14a)....  ...... ....

. .    an1 an2 · · · annfnxnso that the system of equations is transformed to the form   b11 b12 · · · b1n   g1  x1  0 b22 · · · b2n    g2  x2[B]{x} = {g} ⇒ =..  ..... 0.0. .   ..    000 bnnxngn(C.14b)465Hutton: Fundamentals ofFinite Element Analysis466Back MatterAPPENDIX CAppendix C: SolutionTechniques for LinearAlgebraic Equations© The McGraw−HillCompanies, 2004Solution Techniques for Linear Algebraic EquationsIn Equation C.14b, the original coefficient matrix has been transformed to uppertriangular form as all elements below the main diagonal are 0. In this form, thesolution for xn is simply gn /bnn and the remaining values xi are obtained by successive back substitution into the remaining equations.The Gauss method is readily amenable to computer implementation, as described by the following algorithm. For the general form of Equation C.13, wefirst wish to eliminate x1 from the second through nth equations.

To accomplishthis task, we must perform row operations such that the coefficient matrix element ai1 = 0, i = 2, n. Selecting a11 as the pivot element, we can multiply thefirst row by a21 /a11 and subtract the result from the second row to obtain(1)a21 = a21 − a11(1)a22 = a22 − a12...(1)(1)2a21a11(C.15)a2n = a2n − a1nfa21=0a11= f2 − f1a21a11a21a11In these relations, the superscript is used to indicate that the results are from operation on the first column. The same procedure is used to eliminate x1 from theremaining equations; that is, multiply the first equation by ai1 /a11 and subtractthe result from the ith equation. (Note that, if ai1 is 0, no operation is required.)The procedure results in(1)ai1 = 0(1)ai jf(1)ii = 2, nai1= ai j − a1 ji = 2, na11ai1= fi − f1i = 2, na11j = 2, n(C.16)The result of the operations using a11 as the pivot element are represented symbolically asa11 a12 · · · a1n   f1 x1(1)(1)  (1)  0 a22···afx2n 22=(C.17)..........

 ..  0.. .(1)xn(1) (1) 0 an2· · · annf nand variable x1 has been eliminated from all but the first equation. The procedure(1)as the pivot element and the operationsnext takes (newly calculated) element a22Hutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix C: SolutionTechniques for LinearAlgebraic Equations© The McGraw−HillCompanies, 2004C.3 LU Decomposition(1)are repeated so that all elements in the second column below a22become 0.

Carrying out the computations, using each successive diagonal element as the pivotelement, transforms the system of equations to the form of Equation C.14. Thesolution is then obtained, as noted, by back substitutiongnxn =bnnx n−1 =1bn−1,n−1(gn−1 − bn−1,n x n )...(C.18)n1xi =gi −bi j x jbiij=i+1The Gauss elimination procedure is easily programmed using array storageand looping functions (DO loops), and it is much more efficient than invertingthe coefficient matrix. If the coefficient matrix is symmetric (common to manyfinite element formulations), storage requirements for the matrix can be reducedconsiderably, and the Gauss elimination algorithm is also simplified.C.3 LU DECOMPOSITIONAnother efficient method for solving systems of linear equations is the so-calledLU decomposition method.

In this method, a system of linear algebraic equations, as in Equation C.14, are to be solved. The procedure is to decompose thecoefficient matrix [A] into two components [L] and [U ] so thatU11 U12 · · · U1n0 ··· 0L 11 L 21 L 22 · · · 0  0 U22 · · · U2n [A] = [L][U ] =  ..(C.19)....  ...... .... ........ L n1 L n2 · · · L nn0· · · · · · UnnHence, [L] is a lower triangular matrix and [U] is an upper triangular matrix.Here, we assume that [A] is a known n × n square matrix. Expansion of Equation C.19 shows that we have a system of equations with a greater number ofunknowns than the number of equations, so the decomposition into the LU representation is not well defined.

In the LU method, the diagonal elements of [L]must have unity value, so that10 ··· 0 L 211 ··· 0[L] =  ..(C.20)..... .. .. .L n1 L n2 · · · 1467Hutton: Fundamentals ofFinite Element Analysis468Back MatterAPPENDIX CAppendix C: SolutionTechniques for LinearAlgebraic Equations© The McGraw−HillCompanies, 2004Solution Techniques for Linear Algebraic EquationsFor illustration, we assume a 3 × 3 system and write a11 a12 a1310 0U11 U12 U13 a21 a22 a23  =  L 211 0  0 U22 U23 a31 a32 a33L 31 L 32 100 U33(C.21)Matrix Equation C.21 represents these nine equations:a11 = U11a12 = U12a21 = L 21 U11a22 = L 21 U12 + U22a13 = U13(C.22)a31 = L 31 U11a32 = L 31 U12 + L 32 U22a23 = L 21 U13 + U23a33 = L 31 U13 + L 32 U23 + U33Equation C.22 is written in a sequence such that, at each step, only a single unknown appears in the equation.

We rewrite the coefficient matrix [A] and dividethe matrix into “zones” as1 23  a11 哸a12 哸a13a21 a22 哸a23 [A] =  哹哹a哹哹哹哹31 a32 a33(C.23)With reference to Equation C.22, we observe that the first equation correspondsto zone 1, the next three equations represent zone 2, and the last five equationsrepresent zone 3. In each zone, the equations include only the elements of [ A]that are in the zone and only elements of [L ] and [U ] from previous zones andthe current zone.

Hence, the LU decomposition procedure described here is alsoknown as an active zone method.For a system of n equations, the procedure is readily generalized to obtainthe following resultsU 1i = a1ii = 1, nL ii = 1L i1 =ai1U 11i = 2, n(C.24)(C.25)Hutton: Fundamentals ofFinite Element AnalysisBack MatterAppendix C: SolutionTechniques for LinearAlgebraic Equations© The McGraw−HillCompanies, 2004C.3 LU DecompositionThe remaining terms obtained from active zone i, with i ranging from 2 to n, areai j −Lij =U ji = a ji −j−1L im U m jm=1i = 2, nUj jj−1j = 2, 3, 4, . . .

, i − 1 i = j (C.26)L jm U mim=1U ii = aii −i−1i = 2, nL im U mi(C.27)m=1Thus, the decomposition procedure is straightforward and readily amenable tocomputer implementation.Now that the decomposition procedure has been developed, we return to thetask of solving the equations. As we now have the equations expressed in theform of the triangular matrices [L ] and [U ] as[L ][U ]{x } = { f }(C.28)[U ]{x } = {z}(C.29)we see that the productis an n × 1 column matrix, so Equation C.28 can be expressed as[L ]{z} = { f }(C.30)and owing to the triangular structure of [L], the solution for Equation C.30 isobtained easily as (in order)z1 = f1zi = fi −i−1Lij zji = 2, n(C.31)j=1Formation of the intermediate solutions, represented by Equation C.31, is generally referred to as the forward sweep.With the zi value known from Equation C.31, the solutions for the originalunknowns are obtained via Equation C.29 asznxn =U nn(C.32)n1xi =zi −Ui j x jU iij=i+1The process of solution represented by Equation C.32 is known as the backwardsweep or back substitution.469Hutton: Fundamentals ofFinite Element Analysis470Back MatterAppendix C: SolutionTechniques for LinearAlgebraic EquationsAPPENDIX C© The McGraw−HillCompanies, 2004Solution Techniques for Linear Algebraic EquationsIn the LU method, the major computational time is expended in decomposing the coefficient matrix into the triangular forms.

However, this step need beaccomplished only once, after which the forward sweep and back substitutionprocesses can be applied to any number of different right-hand forcing functions{ f } . Further, if the coefficient matrix is symmetric and banded (as is most oftenthe case in finite element analysis), the method can be quite efficient.C.4 FRONTAL SOLUTIONThe frontal solution method (also known as the wave front solution) is an especially efficient method for solving finite element equations, since the coefficientmatrix (the stiffness matrix) is generally symmetric and banded.

In the frontalmethod, assembly of the system stiffness matrix is combined with the solutionphase. The method results in a considerable reduction in computer memory requirements, especially for large models.The technique is described with reference to Figure C.1, which shows anassemblage of one-dimensional bar elements. For this simple example, we knowthat the system equations are of the form   F K 11 K 12U  0000  1 1  K 12 K 22 K 23000UF 2 2   0KKK00UF23333433=(C.33) 00K 34 K 44 K 450 U4 F4  000K 45 K 55 K 56 U  F  5 5 0000K 56 K 66U6F6Clearly, the stiffness matrix is banded and sparse (many zero-valued terms).

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