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5/1/ Finite Difference Method Civil Engineering Majors Authors: Autar Kaw, Charlie Barker Transforming Numerical Methods Education for STEM Undergraduates

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Finite Difference Method

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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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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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Solution Step 1 At node Step 2 At node Step 3 At node

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Solution Cont Step 4 At node Step 5 At node Step 6 At node

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Solving system of equations

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Solution Cont

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Solution Cont Using the approximation of and Gives you

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Solution Cont Step 1 At node Step 2 At node Step 3 At node

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Solution Cont Step 4 At node Step 5 At node Step 6 At node

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Solving system of equations

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Solution Cont

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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| ×10 − ×10 − ×10 − ×10 − ×10 − ×10 − ×10 − ×10 −

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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 ence_method.html

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