Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab (Moukalled F., Mangani L., Darwish M. The finite volume method in computational fluid dynamics. An advanced introduction with OpenFOAM and Matlab.pdf), страница 9

. . þ a1N /N ¼ b1a21 /1 þ a22 /2 þ a23 /3 þ . . . þ a2N /N ¼ b2a31 /1 þ a32 /2 þ a33 /3 þ . . . þ a3N /N ¼ b3...............ð2:41ÞaN1 /1 þ aN2 /2 þ aN3 /3 þ . . . þ aNN /N ¼ bNIn matrix notation, this system of equations is equivalent to2a116 a2166 a316 .6 .6 .6 .4 ..aN1a12a22a32......aN2a13a23a33......aN3......323 2 3/1b1 a1N a2N 76 /2 7 6 b2 7767 6 7 a3N 76 /3 7 6 b3 7766 7.. 76 .. 7..7¼6 .

7. 76 . 7 6 .. 7.6 . 7 6 . 7.. 7... 54 .. 5 4 .. 5.bN aNN/Nð2:42Þor in compact form asA/ ¼ bð2:43Þ2.3.3 The Determinant of a Square MatrixA determinant is a value associated with a square matrix A that can be computedfrom the elements of the matrix through a mathematical procedure and is denoted bydetðAÞ or jAj (which should not be confused with the absolute value notation) [13].242 Review of Vector CalculusThe calculation of the determinant of a matrix of order 2 is straightforward and isthe product of the elements in the main diagonal minus the product of the elementsin the cross diagonal. If A is a square matrix of order 2 then,A¼a11a21a12a22) detðAÞ ¼ a11 a22 a21 a12ð2:44ÞFor higher order matrices the procedure is more involved and is based on thenotion of minors and cofactors.A minor ðmiÞij for an element aij is the determinant that results when the ith rowand jth column are deleted. The cofactor ðcoÞij of an element aij is the value of theminor multiplied by either a positive or a negative sign depending on whetherði þ jÞ is even or odd, respectively.

The mathematical relation between cofactorsand minors can be written asðcoÞij ¼ ð1Þiþj ðmiÞijð2:45ÞThe determinant of a square matrix A of order N is computed by finding thecofactors of one of its rows or its columns, multiplying each cofactor by thecorresponding element, and adding the results. Mathematically this is given by8NP>>>aij ðcoÞij>>< i¼1detðAÞ ¼ or>N>P>>>: aij ðcoÞijfor any jð2:46Þfor any ij¼1It should be clarified that the calculation of the cofactors may require furtherdecomposition of the minor determinants. This decomposition may give rise tofurther decompositions until a determinant with a size of 2 is reached.

Moreover,based on the above discussion it is easily demonstrated that the determinant of anupper, a lower, or a diagonal matrix A of order N is the product of the elementsNQalong its main diagonal, i.e., detðAÞ ¼aii .i¼1Example 6Calculate the determinant of matrix A of order 4 given by2161A¼642402311025305770532.3 Matrices and Matrix Operations25SolutionAs mentioned above, the determinant can be calculated based on the cofactorsof any selected row or column.

A smart choice would be a row or a columnwith the largest number of zeros. Therefore computations will be reduced byselecting either the first row or the last column. The determinant will becalculated using both to further show that the end results will be the same.The signs of cofactors are23þ þ 6 þ þ7674þ þ 5 þ þThe determinant using cofactors of row 1 is computed as 2 0 5 1 detðAÞ ¼ 1 ðcoÞ11 þ 1 ðcoÞ13 ¼ 3 2 0 þ 2 1 5 3 42 5 3 0 1 3The first new determinant is calculated using the cofactors of row 1 while thesecond determinant is calculated using cofactors of column 3 as 2 0 þ 5 3 2 þ 5 2 3 þ 3 1 2 detðAÞ ¼ 22 34 11 55 3¼ 2ð6 0Þ þ 5ð15 þ 2Þ þ 5ð2 12Þ þ 3ð3 4Þ¼ 12 65 50 3detðAÞ ¼ 130The determinant using cofactors of column 4 is calculated as1 01 0 1 detðAÞ ¼ 5 ðcoÞ24 þ 3 ðcoÞ44 ¼ 5 2 3 2 þ 3 1 22 3 4 1 5 1 0 2 The first and second new determinants are calculated using the cofactors ofrow 1 as 3 2 þ 5 2 3 þ 3 2 0 þ 3 1 2 detðAÞ ¼ 52 33 24 11 5¼ 5ð15 þ 2Þ þ 5ð2 12Þ þ 3ð4 0Þ þ 3ð3 4Þ¼ 65 50 12 3detðAÞ ¼ 130As expected, the same value is obtained.262 Review of Vector Calculus2.3.4 Eigenvectors and EigenvaluesConsider a square matrix A and a vector v.

The vector v is an eigenvector of A ifthe product Av results in a vector that has the same direction as v [14–19].Therefore an eigenvector of a matrix is a nonzero vector that does not rotate when isapplied to it. As shown in Fig. 2.11, the only effects may be to change its lengthand/or reverse its direction. Thus, there exist a scalar k such that Av ¼ kv. Thevalue of k is an eigenvalue of A. It is clear that for any constant a the vector av isalso an eigenvector of A because AðavÞ ¼ aAv ¼ akv ¼ kðavÞ.

Thus, a scaledeigenvector is also an eigenvector.If A is symmetric of order N, then it can be shown that A has a set of linearlyindependent eigenvectors denoted v1 ; v2 ; v3 ; . . .; vN . As proved above this set isnot unique. However the corresponding set of their eigenvalues, denotedk1 ; k2 ; k3 ; . . .; kN , which may or may not be equal to each other, is unique. Theeigenvalues of the identity matrix are all ones, and every nonzero vector is aneigenvector of I.In general the eigenvalues of a square matrix A of order N are obtained fromsolving the following equation:Av ¼ kv ) Av ¼ kIv ) ðA kIÞv ¼ 0ð2:47ÞSince, by definition, eigenvectors are nonzero, thenA kI ¼ 0 ) detðA kIÞ ¼ 0ð2:48ÞThe expanded form of Eq. (2.48) is given by2a11 k6 a2166 a316..det66.6..4.aN1a12a22 ka32......aN2va13a23a33 k......aN3............Av3a1Na2N 77a3N 77..7¼07.7..5.aNN kAvFig.

2.11 Effects of multiplying a matrix A by one of its Eigenvectors vð2:49Þ2.3 Matrices and Matrix Operations27As an example, the eigenvalues of the following square matrix of order 2 arefound as:A¼38k 31 ¼ 0 ) ðk 3Þðk 1Þ 8 ¼ 0)8k 111) k 4k 5 ¼ 0 ) ðk þ 1Þðk 5Þ ¼ 0 ) k1 ¼ 1 or k2 ¼ 522.3.5 A Symmetric Positive-Definite Matrix A symmetric matrix A ¼ aij of order N is positive-definite if for all columnvectors p in RN the following inequality holds:pT Ap [ 0ð2:50ÞFor example, if A is an order 3 symmetric matrix given by25A ¼ 43137431458then Eq.

(2.50) for any column vector p of order 3 gives256pT Ap ¼ ½a b c4 3132 3a3 176 77 4 54 b 5c4 82¼ 3ða þ bÞ þ ða þ cÞ2 þ 4ðb þ cÞ2 þ a2 þ 4b2 þ 3c2 [ 0which is positive-definite.If A is a symmetric positive-definite matrix given by2a116 a2166 a316A ¼ 6 ..6 .6 .4 ..aN1a12a22a32......aN2a13a23a33......aN3............3a1Na2N 77a3N 7.. 77. 7.. 7. 5aNNð2:51Þ282 Review of Vector Calculusthen, among others, the following properties apply:1. Any sub-matrix P of A of order M ð1 M N Þ of the form32a11 a12 a1M6 a21 a22 a2M 776P ¼ 6 ....

7....4 .. 5..aM1 aM2 aMMð2:52Þis also positive-definite.2. The N eigenvalues of A, λ1, λ2, λ3,..., λN are positive.3. If all the eigenvalues of a matrix A are positive, then A is positive-definite.4. A has a unique decomposition of the form A ¼ LLT , where L is a lowertriangular matrix.

This decomposition is known as the Cholesky decomposition.2.3.6 Additional Matrix OperationsIf s1 and s2 are scalar functions, I an identity matrix, and A; B, and C are matrices,then the various matrix operations, addition, subtraction, scalar multiplication, andmatrix multiplication, have the following properties listed without proof:A þ ðB þ CÞ ¼ ðA þ BÞ þ Cð2:53ÞAþB¼BþAð2:54Þs1 ðA þ BÞ ¼ s1 A þ s1 Bð2:55Þðs1 þ s2 ÞA ¼ s1 A þ s2 Að2:56ÞAðBCÞ ¼ ðABÞCð2:57ÞAI ¼ IA ¼ Að2:58ÞAðB þ CÞ ¼ AB þ ACð2:59ÞðA þ BÞC ¼ AC þ BCð2:60ÞðA þ BÞT ¼ AT þ BTð2:61Þðs1 AÞT ¼ s1 ATð2:62ÞðABÞT ¼ BT ATð2:63ÞðABÞ1 ¼ B1 A1ð2:64Þ2.4 Tensors and Tensor Operations292.4 Tensors and Tensor OperationsTensors can be thought of as extensions to the ideas already used when definingquantities like scalars and vectors [2, 20, 21].

A scalar is a tensor of rank zero, and avector is a tensor of rank one. Tensors of higher rank (2, 3, etc.) can be developedand their main use is to manipulate and transform sets of equations. Since within thescope of this book only tensors of rank two are needed, they will be referred tosimply as tensors.Similar to the flow velocity vector v, the deviatoric stress tensor s (Fig. 2.12) willbe referred to frequently in this book and is used here to illustrate tensor operations.Let x; y; and z represent the directions in an orthonormal Cartesian coordinatesystem, then the stress tensor s and its transpose designated with superscript TðsT Þare represented in terms of their components as2sxxs ¼ 4 syxszxsxysyyszy32sxxsxzsyz 5 sT ¼ 4 sxyszzsxz3szxszy 5szzsyxsyysyzð2:65ÞSimilar to writing a vector in terms of its components, defining the unit vectors i, j,and k in the x; y; and z direction, respectively, the tensor s given by Eq.

(2.65) can bewritten in terms of its components ass ¼ iisxx þ ijsxy þ iksxz þ jisyx þ jjsyy þ jksyz þ kiszx þ kjszy þ kkszzð2:66ÞEquation (2.66) allows defining a third type of vector product for multiplyingtwo vectors, known as the dyadic product, and resulting in a tensor with its components formed by ordered pairs of the two vectors. In specific, the dyadic productFig. 2.12 Schematic of astress tensor fieldyyyxyzzyxyzxzzxxxz302 Review of Vector Calculusof a vector v by itself, arising in the formulation of the momentum equation of fluidflow, gives9fvvg ¼ ðui þ vj þ wkÞðui þ vj þ wkÞ >2>>uu=¼ iiuu þ ijuv þ ikuw þ) fvvg ¼ 4 vu>jivu þ jjvv þ jkvw þ>wu>;kiwu þ kjwv þ kkwwuvvvwv3uwvw 5wwð2:67ÞThe gradient of a vector v is a tensor given by9@@@>i þ j þ k ðui þ vj þ wkÞ >>>2@x@y@z>@u>>>>>6@u@v@w>>6 @xþ¼ ii þ ij þ ik=6 @u@x@x@x) frvg ¼ 66 @y>@u@v@w>6>>þji þ jj þ jk4 @u>>@y@y@y>>>>@z>@u@v@w>;ki þ kj þ kk@z@z@zfrvg ¼@v@x@v@y@v@z3@w@x 77@w 77@y 77@w 5@zð2:68ÞThe sum of two tensors r and s is a tensor R whose components are the sum ofthe corresponding components of the two tensors, i.e.,2rxx þ sxxR ¼ r þ s ¼ 4 ryx þ syxrzx þ szx3rxz þ sxzryz þ syz 5rzz þ szzrxy þ sxyryy þ syyrzy þ szyð2:69ÞMultiplying a tensor s by a scalar s results in a tensor whose components aremultiplied by that scalar, i.e.,2ssxxfssg ¼ 4 ssyxsszxssxyssyysszy3ssxzssyz 5sszzð2:70ÞThe dot product of a tensor s by a vector v results in the following vector:½ s v ¼iisxx þ ijsxy þ iksxz þ jisyx þjjsyy þ jksyz þ kiszx þ kjszy þ kkszz ðui þ vj þ wkÞð2:71Þ2.4 Tensors and Tensor Operations31which upon expanding becomes½s v ¼ ii isxx u þ ii jsxx v þ ii ksxx w þ ij isxy u þ ij jsxy vþ ij ksxy w þ ik isxz u þ ik jsxz v þ ik ksxz w þ ji isyx uþ ji jsyx v þ ji ksyx w þ jj isyy u þ jj jsyy v þ jj ksyy wþ jk isyz u þ jk jsyz v þ jk ksyz w þ ki iszx u þ ki jszx vþ ki kszx w þ kj iszy u þ kj jszy v þ kj kszy w þ kk iszz uþ kk jszz v þ kk kszz wð2:72ÞUsing Eq.