fundamental-engineering-optimization-methods.pdf

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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 107 Linear Programming Methods The solution is obtained by solving the FONC, the constraint in (5.30), for x as: ࢞ሺ࢜ሻ ൌ െࡽ ିଵ ሺ࡭ ࢜ ൅ ࢉሻ (5.32) and substituting it in the Lagrangian function to obtain: Ȱሺ࢜ሻ ൌ െ ሺ࡭ ࢜ ൅ ࢉሻ ିଵ ሺ࡭ ࢜ ൅ ࢉሻ െ ࢜ ൌ െ ሺ࡭ࡽ ିଵ ሻ࢜ െ ሺࢉ ିଵ ൅ ࢈ ሻ࢜ െ ିଵ (5.33) In terms of the dual function, the dual QP problem is defined as: ࢜ஹ૙ Ȱሺ࢜ሻ ൌ െ ሺ࡭ ࢜ ൅ ࢉሻ ିଵ ሺ࡭ ࢜ ൅ ࢉሻ െ ࢜ (5.34) The dual problem can also be solved by application of FONC, where the gradient and Hessian of Ȱሺ࢜ሻ are given as: ׏Ȱ ൌ െ࡭ࡽ ିଵ ሺ࡭ ࢜ ൅ ࢉሻ െ ࢈ǡ ׏ Ȱ ൌ െ࡭ࡽ ିଵ (5.35) By solving ׏ Ȱ ൌ Ͳǡ we obtain the solution to the Lagrange multipliers as: ࢜ ൌ െሺ࡭ࡽ ିଵ ିଵ ሺ࡭ ିଵ ࢉ ൅ ࢈ሻ (5.36) where the non-negativity of v is implied. Finally, the solution to the design variables is obtained from (5.32) as: ࢞ ൌ ࡽ ିଵ ሺ࡭ࡽ ିଵ ିଵ ሺ࡭ ିଵ ࢉ ൅ ࢈ሻ െࡽ ିଵ (5.37) The dual methods have been successfully applied in structural mechanics. As an example of the dual QP problem, we consider a one-dimensional finite element analysis (FEA) problem involving two nodes. Example 5.10: Finite Element Analysis (Belegundu and Chandrupatla, p. 187) Let ݍ ǡ ݍ represent nodal displacements in the simplified two node structure, and assume that a load P , where ܲ ൌ ͸Ͳ݇ܰǡ is applied at node 1. The FEA problem is formulated as minimization of the potential energy function given as: ς ൌ ͳ ʹ ࡷࢗ െ ࢗ Subject to: ݍ ൑ ͳǤʹ In the above problem, ൌ ሾݍ ǡ ݍ represents the vector of nodal displacements.
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Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 108 Linear Programming Methods The stiffness matrix K for the problem is given as: ࡷ ൌ ଵ଴ ʹ െͳ െͳ ͳ Ǥ For this problem: ࡽ ൌ ࡷǡ ࢌ ൌ ሾܲǡ Ͳሿ ǡ ࢉ ൌ െࢌǡ ࡭ ൌ ሾ Ͳ ͳ ሿǡ ࢈ ൌ ͳǤʹǤ Further, ࡭ࡽ ିଵ ൌ ͸ ൈ ͳͲ ିହ ǡ ࢉ ିଵ ൌ െͳǤͺǡ ࢉ ିଵ ࢉ ൌ ͳǤͲͺ ൈ ͳͲ ିହ Ǥ We use (5.33) to obtain the dual function as: Ȱሺ࢜ሻ ൌ െ͵ ൈ ͳͲ ିହ ݒ െ ͲǤ͸ݒ െ ͳǤͲͺ ൈ ͳͲ ିହ ² From (5.36) the solution to Lagrange multiplier is: ݒ ൌ ͳ ൈ ͳͲ ² Then, from (5.37), the optimum solution to the design variables is: ݍ ൌ ͳǤͷ ݉݉ǡ ݍ ൌ ͳǤʹ ݉݉Ǥ The optimum value of potential energy function is: ς ൌ ͳʹͻ ܰ݉Ǥ Next, we proceed to define and solve the Linear Complementarity Problem.
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