numerical solution of finite element equations pdf

... about the numerical solution of stochastic differential equations on the webpage of the first author under Numerical Methods”: http://www.business.uts.edu.au/ finance/staff/Eckhard /Numerical Methods.html ... collection of events, sigma-algebra A filtration E(X) expectation of X E(X | A) conditional expectation of X under A P (A) probability of A P (A | B) probability of A conditioned on B ∈ element of ∈ ... the existence and uniqueness of solutions of SDEs These tools and results provide the basis for the application and numerical solution of stochastic differential equations with jumps 1.1 Stochastic...

Ngày tải lên: 19/02/2014, 22:20

... about the numerical solution of stochastic differential equations on the webpage of the first author under Numerical Methods”: http://www.business.uts.edu.au/ finance/staff/Eckhard /Numerical Methods.html ... collection of events, sigma-algebra A filtration E(X) expectation of X E(X | A) conditional expectation of X under A P (A) probability of A P (A | B) probability of A conditioned on B ∈ element of ∈ ... the existence and uniqueness of solutions of SDEs These tools and results provide the basis for the application and numerical solution of stochastic differential equations with jumps 1.1 Stochastic...

Ngày tải lên: 28/06/2014, 17:20

... we consider the development of numerical software for the solution of such problems Our discussion opens with a brief introduction to the theory of delay-differential equations Attention is paid ... Delay-differential equations In this thesis we are concerned with the numerical solution of delay-differential equations (DDE's) Delay-differential equations may best be regarded as extensions of ordinary ... under Prof C T H Baker in the department of mathematics at the university of Manchester In 1986 he obtained the degree of MSc in Numerical Analysis and Computation and held the position of temporary...

Ngày tải lên: 11/10/2014, 04:02

... of the root location algorithm discussed in (2.7) Bibliography 415 418 291 appendix A a formal proof for (1.1.3:14) 292 A formal proof for (1.1.3:14) In this appendix we give a formal proof of ... transitive closure of E Heuristically, by an extension of the above arguments, we may think of Ei as defining the set of points whose continuity bounds can affect the continuity bound, ci , of yi El ... the notation of (5), s q Es q+ whereas t q ,Et 1+ Thus k' < k cannot be true and the result then follows [] We can now use this result to prove part (ii) of Proposition 1: PROOF of Proposition...

Ngày tải lên: 11/10/2014, 04:03

... (2.2.4) is called backsubstitution The combination of Gaussian elimination and backsubstitution yields a solution to the set of equations The advantage of Gaussian elimination and backsubstitution ... Analysis of Numerical Methods (New York: Wiley), §2.1 Johnson, L.W., and Riess, R.D 1982, Numerical Analysis, 2nd ed (Reading, MA: AddisonWesley), §2.2.1 Westlake, J.R 1968, A Handbook of Numerical ... Inversion and Solution of Linear Equations (New York: Wiley) Suppose we are able to write the matrix A as a product of two matrices, L·U=A (2.3.1) where L is lower triangular (has elements only...

Ngày tải lên: 24/12/2013, 12:16

... from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software ... “7/8”; it is that factor at each hierarchical level of the recursion In total it reduces the process of matrix multiplication to order N log2 instead of N What about all the extra additions in (2.11.3)–(2.11.4)? ... to trade@cup.cam.ac.uk (outside North America) c22 = Q1 + Q3 − Q2 + Q6 104 Chapter Solution of Linear Algebraic Equations CITED REFERENCES AND FURTHER READING: Strassen, V 1969, Numerische Mathematik,...

Ngày tải lên: 24/12/2013, 12:16

... be linear) Modeling of the ring test Finite element modeling of the ring specimens using ANSYS software were modeled by cubic threedimensional as shown in Fig The elements of steel ring and concrete ... strain In the finite element analysis the strain information was used as input using the assumption of a constrained radial displacement of the nodes of the outer circumference of the steel ring ... elements will start cracking as the stress in a particular layer exceeds the tensile strength of concrete (f’ t) The ring specimens The FEM of the ring test was performed using the software of...

Ngày tải lên: 21/07/2014, 17:21

... Fundamentals of Finite Element Analysis Basic Concepts of the Finite Element Method Text 1.4 Brief History of the Finite Element Method compute additional, derived variables, such as reaction forces, element ... Hutton: Fundamentals of Finite Element Analysis Basic Concepts of the Finite Element Method Text © The McGraw−Hill Companies, 2004 1.2 How Does the Finite Element Method Work? of the cylinder (Figure ... equal-length finite elements The areas are average over the respective length of cylinder (element length ϭ L͞4) Hutton: Fundamentals of Finite Element Analysis Basic Concepts of the Finite Element Method...

Ngày tải lên: 08/08/2014, 17:20

... 24 14 13 12 29 11 10 Figure 8.8 Lines of constant velocity potential ␾ for the finite element solution of Example 8.2 311 Hutton: Fundamentals of Finite Element Analysis 312 Applications in Fluid ... in Fluid Mechanics 15 16 14 Hutton: Fundamentals of Finite Element Analysis © The McGraw−Hill Companies, 2004 309 Hutton: Fundamentals of Finite Element Analysis 310 Applications in Fluid Mechanics ... similarity of the two problems is further illustrated by the finite element solution of the previous example using the velocity potential function EXAMPLE 8.2 Obtain a finite element solution for...

Ngày tải lên: 08/08/2014, 17:20

... set of equations to be solved can be written as the N ×N set of equations (AT · A) · x = (AT · b) (2.0.4) where AT denotes the transpose of the matrix A Equations (2.0.4) are called the normal equations ... event, the solution space consists of a particular solution xp added to any linear combination of (typically) N − M vectors (which are said to be in the nullspace of the matrix A) The task of finding ... (bj = all zero elements except for in the jth component) The corresponding x’s are then the columns of the matrix inverse of A (§2.1 and §2.3) • Calculation of the determinant of a square matrix...

Ngày tải lên: 15/12/2013, 04:15

... Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) Ralston, A., and Rabinowitz, P 1978, A First Course in Numerical Analysis, 2nd ed (New ... the same linear combination of the rows of the b’s and (which then is no longer the identity matrix, of course) • Interchanging any two columns of A gives the same solution set only if we simultaneously ... interchange corresponding rows of the x’s and of Y In other words, this interchange scrambles the order of the rows in the solution If we this, we will need to unscramble the solution by restoring the...

Ngày tải lên: 15/12/2013, 04:15

... zero all elements in a column of the matrix situated below a chosen element Thus we arrange for the first Householder matrix Q1 to zero all elements in the first column of A below the first element ... first element Similarly Q2 zeroes all elements in the second column below the second element, and so on up to Qn−1 Thus 100 Chapter Solution of Linear Algebraic Equations for (j=1;j

Ngày tải lên: 15/12/2013, 04:15

... Computer Solution of Linear Algebraic Systems (Englewood Cliffs, NJ: Prentice-Hall), Chapters 9, 16, and 18 Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations ... modify the loop of the above fragment and (e.g.) divide by powers of ten, to keep track of the scale separately, or (e.g.) accumulate the sum of logarithms of the absolute values of the factors ... Press) 2.4 Tridiagonal and Band Diagonal Systems of Equations The special case of a system of linear equations that is tridiagonal, that is, has nonzero elements only on the diagonal plus or minus...

Ngày tải lên: 24/12/2013, 12:16

... improved solution x 2.5 Iterative Improvement of a Solution to Linear Equations Obviously it is not easy to obtain greater precision for the solution of a linear set than the precision of your ... follows: The diagonal elements are in a[1 n][m1+1] Subdiagonal elements are in a[j n][1 m1] (with j > appropriate to the number of elements on each subdiagonal) Superdiagonal elements are in a[1 ... limitations of bandec, and the above routine does take advantage of the opportunity In general, when TINY is returned as a diagonal element of U , then the original matrix (perhaps as modified by roundoff...

Ngày tải lên: 24/12/2013, 12:16

... inverse of the matrix A, so that B0 · A is approximately the identity matrix Define the residual matrix R of B0 as 58 Chapter Solution of Linear Algebraic Equations We can define the norm of a matrix ... discussion of the use of SVD in this application to Chapter 15, whose subject is the parametric modeling of data SVD methods are based on the following theorem of linear algebra, whose proof is beyond ... wrong solution to get an improved solution An important extra benefit occurs if we obtained the original solution by LU decomposition In this case we already have the LU decomposed form of A,...

Ngày tải lên: 24/12/2013, 12:16

... same permutation of the columns of U, elements of W, and columns of V (or rows of VT ), or (ii) forming linear combinations of any columns of U and V whose corresponding elements of W happen to ... Value Decomposition A A⋅x = b (a) null space of A solutions of A⋅x = d solutions of A ⋅ x = c′ SVD solution of A ⋅ x = c range of A d c′ c SVD solution of A⋅x = d (b) Figure 2.6.1 (a) A nonsingular ... combination of the set of equations that we are trying to solve The resolution of the paradox is that we are throwing away precisely a combination of equations that is so corrupted by roundoff error...

Ngày tải lên: 24/12/2013, 12:16

... • Each of the first N locations of ija stores the index of the array sa that contains the first off-diagonal element of the corresponding row of the matrix (If there are no off-diagonal elements ... the last off-diagonal element of the last row (It can be read to determine the number of nonzero elements in the matrix, or the number of elements in the arrays sa and ija.) Location N + of sa is ... sa of the most recently stored element of a previous row.) • Location of ija is always equal to N + (It can be read to determine N ) • Location N + of ija is one greater than the index in sa of...

Ngày tải lên: 24/12/2013, 12:16

... forms] Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and Statistics (New ... square root” of the matrix A The components of LT are of course related to those of L by LT = Lji ij (2.9.3) Writing out equation (2.9.2) in components, one readily obtains the analogs of equations ... [4] developed an algorithm for fast solution of the symmetric Toeplitz problem, by a bordering method, that is, Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)...

Ngày tải lên: 21/01/2014, 18:20

... forms] Westlake, J.R 1968, A Handbook of Numerical Matrix Inversion and Solution of Linear Equations (New York: Wiley) [2] von Mises, R 1964, Mathematical Theory of Probability and Statistics (New ... square root” of the matrix A The components of LT are of course related to those of L by LT = Lji ij (2.9.3) Writing out equation (2.9.2) in components, one readily obtains the analogs of equations ... America) In §2.4 the case of a tridiagonal matrix was treated specially, because that particular type of linear system admits a solution in only of order N operations, rather than of order N for the...

Ngày tải lên: 21/01/2014, 18:20

Ngày tải lên: 29/03/2014, 02:20