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Topics to be covered in this course--not
necessarily in this order.
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Motivating examples of calculus of variations
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Mathematical preliminaries: normed vectors spaces, functionals (continuous
and linear), directional derivative, concept of variation, Gateaux
variation, Frechet
differential, etc.
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Fundamental lemmas of calculus of variation
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Euler-Lagrange (E-L) equations
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Applications of E-L equation
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Extensions of E-L equation to multiple derivatives, independent variables,
multiple state variables
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Isoperimetric problems--global and local (finite subsidiary)
constraints
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Applications of optimizing functionals subject to constraints
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Applications in mechanics: strong and weak forms of governing equations
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Variable end conditions--transversality conditions
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Size optimization of a bar for maximum stiffness
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Self-adjointness and optimization with weak variational form
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Optimization with side constraints (variable bounds)
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Worst load scenario for an axially loaded stiffest bar
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Min-max type problem with stress constraints
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Beam problems for stiffmess and strength
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Optimization of a beam for given deflection
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Variational formulations for the eigenvalue problems: strings, bars, beams, and other elastic strucrures.
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Optimum design of a column
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Variable-thickness optimization of plates
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Sufficient conditions for E-L optimum
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Applications of sufficient conditions
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Finite dimensional optimization--A summary and highlights
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Numerical optimization techniques
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Using the optimization tool-box in Matlab
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Truss topology optimization
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Sensitivity analysis
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Frame topology optimization
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Compliant mechanism design using topology optimization of trusses and
frames
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Topology optimization using continuum elements
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Optimality criteria method
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Shape optimization of structures
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Applications to multi-physics problems
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Material interpolation techniques for topology optimization
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