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5/1/2015 http://numericalmethods.eng.usf.edu 1 Finite Difference Method Civil Engineering Majors Authors: Autar Kaw, Charlie Barker http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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Finite Difference Method http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

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3 Finite Difference Method An example of a boundary value ordinary differential equation is The derivatives in such ordinary differential equation are substituted by finite divided differences approximations, such as

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http://numericalmethods.eng.usf.edu4 Example Take the case of a pressure vessel that is being tested in the laboratory to check its ability to withstand pressure. For a thick pressure vessel of inner radius a and outer radius b, the differential equation for the radial displacement u of a point along the thickness is given by The pressure vessel can be modeled as, Substituting these approximations gives you,

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http://numericalmethods.eng.usf.edu5 Solution Step 1 At node Step 2 At node Step 3 At node

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http://numericalmethods.eng.usf.edu6 Solution Cont Step 4 At node Step 5 At node Step 6 At node

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http://numericalmethods.eng.usf.edu7 Solving system of equations

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http://numericalmethods.eng.usf.edu8 Solution Cont

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http://numericalmethods.eng.usf.edu9 Solution Cont Using the approximation of and Gives you

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http://numericalmethods.eng.usf.edu10 Solution Cont Step 1 At node Step 2 At node Step 3 At node

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http://numericalmethods.eng.usf.edu11 Solution Cont Step 4 At node Step 5 At node Step 6 At node

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http://numericalmethods.eng.usf.edu12 Solving system of equations

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http://numericalmethods.eng.usf.edu13 Solution Cont

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http://numericalmethods.eng.usf.edu14 Comparison of radial displacements Table 1 Comparisons of radial displacements from two methods ru exact u 1st order |єt||єt|u 2nd order |єt||єt| 50.0038731 0.00000.00387310.0000 5.60.00361100.00361651.5160×10 −1 0.00361151.4540×10 −2 6.20.00341520.00342222.0260×10 −1 0.00341591.8765×10 −2 6.80.00326830.00327431.8157×10 −1 0.00326891.6334×10 −2 7.40.00315830.00316181.0903×10 −1 0.00315869.5665×10 −3 80.0030769 0.00000.00307690.0000

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Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/finite_differ ence_method.html

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THE END http://numericalmethods.eng.usf.edu

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Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker

Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker

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