Curve & Surface Fitting by Peter Lancaster

By Peter Lancaster

From the Preface: the advance of a few of the strategies utilized in special effects is determined by quite a lot of mathematical tools for curve and floor becoming. when you consider that entry to pcs calls for little or no education in arithmetic, lots of those tools is probably not simply understood through the good number of those who are now capable of use robust computing gear. the aim of this booklet is to bare to the (but probably mathematically unsophisticated) person the rules and significant positive factors of a number of easy tools for curve and floor becoming which are at the moment in use.

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Example text

Compute the value for the cross-sectional area A = 4bt (here, and for I later, we are using a simplification customary for a thin-walled construction). • Compute the value of the stress σ = P/A: where P is drawn from a database. • Compute the Euler buckling load Pcr = π 2EI/L2 (where I = 2b3t/3). • Compute the applied load over the cross-sectional contour N = P/(4b). • Compute the sidewall buckling load Ncr = 4π 2D/b2 (where D = Et2/12(1 − υ2) is the plate stiffness coefficient). This module also computes the values for the constraints which for our problem are given by: g1 = 1 − t ≤ 0 the minimum gauge constraint t0 σ −1 ≤ 0 the overstress constraint σa P − 1 ≤ 0 the Euler buckling constraint g3 = Pcr N g4 = − 1 ≤ 0 the wall plate buckling constraint Ncr g2 = This information is supplied to the search engine which computes the value of the objective function and employs one of the algorithms discussed in later chapters to locate an estimate of 20 Multidisciplinary Design Optimization Supported by Knowledge Based Engineering the optimizing point and the optimizing values for the design variables.

3. Assuming the output from the search process has provided a feasible design point, the objective function is examined to see if is optimal. If the answer to the question is “yes,” then the process stops; if “no,” then the process returns to step 2 with the updated design variables. Although this walkthrough has greatly simplified the process of finding an optimal design, it does cover the main actions undertaken by optimum seeking methods and also, in principle, is applicable to the MDO methods described in Chapters 9 and 10.

It is considered to be active if it is violated, that is, g(x) > 0, or it lies within a narrow band close the edge of the feasible region, that is, g(x) + ε > 0, with ε <<1 if the constraints are normalized to the numerical value 1. It is now possible to see that an equality constraint can be considered as a special case of an inequality constraint which is always active. 4, which discusses characterizing the optimum, equality constraints are explicitly reintroduced into the discussion. 4) and a second where the design variables violate one or more of the constraints.

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