Barna Szabó - Finite Element Analysis

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Finite Element Analysis: краткое содержание, описание и аннотация

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Finite Element Analysis <p><b>An updated and comprehensive review of the theoretical foundation of the finite element method</b> <p>The revised and updated second edition of <i>Finite Element Analysis: Method, Verification, and Validation</i> offers a comprehensive review of the theoretical foundations of the finite element method and highlights the fundamentals of solution verification, validation, and uncertainty quantification. Written by noted experts on the topic, the book covers the theoretical fundamentals as well as the algorithmic structure of the finite element method. The text contains numerous examples and helpful exercises that clearly illustrate the techniques and procedures needed for accurate estimation of the quantities of interest. In addition, the authors describe the technical requirements for the formulation and application of design rules. <p>Designed as an accessible resource, the book has a companion website that contains a solutions manual, PowerPoint slides for instructors, and a link to finite element software. This important text: <ul><li>Offers a comprehensive review of the theoretical foundations of the finite element method</li> <li>Puts the focus on the fundamentals of solution verification, validation, and uncertainty quantification</li> <li>Presents the techniques and procedures of quality assurance in numerical solutions of mathematical problems</li> <li>Contains numerous examples and exercises</li></ul> <p>Written for students in mechanical and civil engineering, analysts seeking professional certification, and applied mathematicians, <i>Finite Element Analysis: Method, Verification, and Validation, Second Edition</i> includes the tools, concepts, techniques, and procedures that help with an understanding of finite element analysis.

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The solution is

(1.125) Using eq 1106we get 1126 where is the differ - фото 788

Using eq. (1.106)we get

(1.126) where is the difference between and its linear interpolant 1 - фото 789

where is the difference between and its linear interpolant 1127 a - фото 790is the difference between and its linear interpolant 1127 and compute Ref - фото 791and its linear interpolant:

(1.127) and compute Referring to Theorem 15 the error in energy norm associated with - фото 792

and compute

Referring to Theorem 15 the error in energy norm associated with the k th - фото 793

Referring to Theorem 1.5, the error in energy norm associated with the k th element is

(1.128) and the relative error in energy norm associated with the k th element is - фото 794

and the relative error in energy norm associated with the k th element is:

(1.129) The error of approximation over the entire domain is 1130 By Theorem 12 - фото 795

The error of approximation over the entire domain is:

(1.130) By Theorem 12 the exact value of the potential energy is 1131 and the - фото 796

By Theorem 1.2, the exact value of the potential energy is

(1.131) and the relative error in energy norm on the entire domain is 1132 - фото 797

and the relative error in energy norm on the entire domain is:

(1.132) Finite Element Analysis - изображение 798

Remark 1.13In estimating the local error we used Finite Element Analysis - изображение 799. It can be shown that in the special case of this problem ( картинка 800) this relationship holds and therefore using the equal sign in eq. (1.128)is justified. In the general case ( Finite Element Analysis - изображение 801) however, Finite Element Analysis - изображение 802and eq. (1.128)will be an estimate of the local error in the finite element solution. Therefore the equal sign in eq. (1.128)has to be replaced by the approximately equal ( картинка 803) sign and the first equal sign in eq. (1.130)has to be replaced with the less or equal ( картинка 804) sign.

Example 1.13This example illustrates the distribution of the relative error among the elements for a fixed mesh and polynomial degree for selected fractional values of α . Uniform mesh on the domain картинка 805with Finite Element Analysis - изображение 806and Finite Element Analysis - изображение 807for Finite Element Analysis - изображение 808is used. The exact solution for картинка 809is shown in Fig. 1.12. The percent relative error in energy norm associated with the k th element, given by eq. (1.129), is shown in Table 1.4and the relative error for the entire domain is shown in the last column.

It is seen that for all values of α the maximum error is associated with the first element.

Example 1.14This example illustrates the distribution of the relative error among the elements for a fixed mesh and polynomial degree for selected integer values of α . Uniform mesh on the domain картинка 810with Finite Element Analysis - изображение 811and Finite Element Analysis - изображение 812for Finite Element Analysis - изображение 813is used. The percent relative error in energy norm associated with the k th element, given by eq. (1.129), is shown in Table 1.5and the relative error for the entire domain is shown in the last column.

The error of approximation for картинка 814is zero. This follows directly from Theorem 1.4: The exact solution is a polynomial of degree 2. Therefore it lies in the finite element space and hence the finite element solution is the same as the exact solution.

Remark 1.14In the foregoing discussion it was tacitly assumed that all data computed by numerical integration were accurate and the coefficient matrices of the linear equations were such that small changes in the right‐hand‐side vector produce small changes in the solution vector. This happens when the condition number of the coefficient matrix is reasonably small. In the finite element method the condition number depends on the choice of the shape functions, the mapping functions and the mesh. In one‐dimensional setting the mapping is linear and the shape functions are energy‐orthogonal, therefore round‐off errors are not significant. This is not the case in two and three dimensions, however.

Table 1.4 Example: Element‐by‐element and total relative errors in energy norm (percent) for selected fractional values of α .

Element number
α 1 2 3 4 5 картинка 815
1.25 79.49 7.50 2.80 1.63 1.12 4.80
1.15 99.52 4.06 1.63 0.97 0.67 3.92
1.05 29.56 1.24 0.53 0.32 0.22 1.77
0.95 18.89 1.16 0.52 0.32 0.22 2.41
0.85 42.94 3.26 1.52 0.94 0.67 9.84
0.75 60.39 5.14 2.47 1.56 1.11 22.37
0.65 76.07 6.86 3.39 2.16 1.56 42.91
0.55 91.80 8.44 4.28 2.76 2.00 76.22

Table 1.5 Example: Element‐by‐element and total relative errors in energy norm (percent) for selected integer values of α .

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