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Rational Approach to Mesh Refinement and Error Analysis

Rational Approach to Mesh Refinement and Error Analysis. Professor J. O. Dow Civil Engineering Dept Sam Semakula. Hard Problem. One of the most difficult problems in Engineering Design is determining stresses and strains in complex parts

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Rational Approach to Mesh Refinement and Error Analysis

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  1. Rational Approach to Mesh Refinement and Error Analysis Professor J. O. Dow Civil Engineering Dept Sam Semakula

  2. Hard Problem • One of the most difficult problems in Engineering Design is determining stresses and strains in complex parts • Numerical Methods used to find approximate solutions to these difficult problems • Finite Element (FE) and Finite Difference (FD) methods

  3. Question? • Does the approximate solution validate the design? • Is there a “cheaper” process that will yield a “good” solution? • Mesh refinement • Rational vs. Heuristic Refinement • Error analysis • Universal error measures

  4. What has been done in terms of error analysis? • Procedures for evaluating errors in the FE and FD methods have been developed • Error evaluated by comparing approximate solution with a continuous smooth solution • Smoothed solution assumed to be more accurate that discontinuous solution • However, smooth solution only used in forming error results in the model

  5. Pass continuous solution through the average of the discontinuities in the FE solution • Error quantified by computing the mean square error

  6. Adaptive Refinement

  7. Problems with Adaptive Refinement • Results from previous mesh models are not used • Refinement of next model is heuristically based • No way to quantify amount of mesh refinement • Refinement dimension

  8. Goals • Develop techniques that use previous mesh results in forming new approximate solution • Quantify solution “gradients” • Develop a correlation between solution “gradients” and desired approximate solutions • Develop point-wise error measures for FE and FD methods • Use MatLab and other FE software to test theory and produce results

  9. Schedule

  10. Thank You

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