Shape Sensitivity Analysis and Design Optimization of Linear, Thermoelastic Solids

Hou, Gene; Sheen, Jeen S.; Chuang, Ching H.

doi:10.2514/6.1990-1012

Cited by 10 publications

(21 citation statements)

References 11 publications

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“…n.r = h on h (4) where, n is the unit normal vector on the boundary . Here, g and h are the subsets of the boundary .…”

Section: Flow Equationsmentioning

confidence: 99%

“…Several techniques have been tried for carrying out aerodynamic shape optimization. Some of the methods are random search methods [1], complex Taylor series expansion approach [2], automatic differentiation method [3], direct differentiation method [4] and adjoint 356 D. N. SRINATH AND S. MITTAL based methods [5]. Among the most commonly used methods are gradient based methods in which a specified objective function is minimized or maximized with respect to design shape parameters.…”

Section: Introductionmentioning

confidence: 99%

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Optimal airfoil shapes for low Reynolds number flows

Srinath

Mittal

2008

Numerical Methods in Fluids

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SUMMARYFlow over NACA 0012 airfoil is studied at = 4 • and 12 • for Re 500. It is seen that the flow is very sensitive to Re. A continuous adjoint based method is formulated and implemented for the design of airfoils at low Reynolds numbers. The airfoil shape is parametrized with a non-uniform rational B-splines (NURBS). Optimization studies are carried out using different objective functions namely: (1) minimize drag, (2) maximize lift, (3) maximize lift to drag ratio, (4) minimize drag and maximize lift and (5) minimize drag at constant lift. The effect of Reynolds number and definition of the objective function on the optimization process is investigated. Very interesting shapes are discovered at low Re. It is found that, for the range of Re studied, none of the objective functions considered show a clear preference with respect to the maximum lift that can be achieved. The five objective functions result in fairly diverse geometries. With the addition of an inverse constraint on the volume of the airfoil the range of optimal shapes, produced by different objective functions, is smaller. The non-monotonic behavior of the objective functions with respect to the design variables is demonstrated. The effect of the number of design parameters on the optimal shapes is studied. As expected, richer design space leads to geometries with better aerodynamic properties. This study demonstrates the need to consider several objective functions to achieve an optimal design when an algorithm that seeks local optima is used.

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“…n.r = h on h (4) where, n is the unit normal vector on the boundary . Here, g and h are the subsets of the boundary .…”

Section: Flow Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal airfoil shapes for low Reynolds number flows

Srinath

Mittal

2008

Numerical Methods in Fluids

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“…(1) Dems and Mroz (2) Meric (1) Dems and Mroz (2) Tortorelli (3) Hou (4) Bobaru and Mukherjee (5) Grindeanu (6) (7) (8)…”

Section: Mericmentioning

confidence: 99%

Shape Optimization of Thermoelastic Fields for Mean Compliance Minimization

Katamine

Yoshioka²,

Matsuura³

et al. 2011

Trans.JSME(C)

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This paper presents a numerical analysis method for shape optimization in order to achieve stiffness maximization in thermoelastic fields. Mean compliance is used as an objective functional for the shape optimization problem. The mean compliance minimization problem on the thermoelastic fields is formulated on volume constraint condition. The shape gradient of the shape optimization problems is derived theoretically using the adjoint variable method, the Lagrange multiplier method and the formulae of the material derivative. Reshaping is accomplished using a traction method that was proposed as a solution to shape optimization problems. In addition, a new numerical procedure for the shape optimization is proposed. The validity of the proposed method is confirmed based on the results of 2D numerical analysis.

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“…The equations and the boundary conditions for the adjoint variables are obtained by setting the expression given in (9) to zero. This leads to…”

Section: Adjoint Equationsmentioning

confidence: 99%

“…Different methods exist to calculate the gradient of the objective function. The complex Taylor series expansion approach [2][3][4][5], and the automatic- [6][7][8] and direct-differentiation [9] method can be employed to calculate the gradient of the objective function. Another possible approach is via a simple finite difference procedure.…”

Section: Introductionmentioning

confidence: 99%

A stabilized finite element method for shape optimization in low Reynolds number flows

Srinath

Mittal

2007

Numerical Methods in Fluids

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SUMMARYA gradient-based optimization procedure based on a continuous adjoint approach is formulated and implemented for steady low Reynolds number flows. A stabilized finite element formulation is proposed to solve the adjoint equations. The accuracy of the gradients from the adjoint approach is verified against the ones computed from a simple finite difference procedure. The validation of the formulation and its implementation is carried out via flow past an elliptical bump whose eccentricity is used as a design parameter. Shape design studies for the elliptical bump are then carried on with a more complex 4th order Bézier parametrization of the bump. Results for, both, optimal design and inverse problems are presented. Using different initial guesses, multiple optimal shapes are obtained. A multi-objective function with additional constraints on the volume and the drag coefficient of the bump is utilized. It is seen that as more constraints are added to the objective function the design space is constrained and the multiple optimal shapes become progressively similar to each other. The study demonstrates the usefulness of this tool in obtaining multiple engineering solutions to a given design problem and also providing a framework to impose multiple constraints simultaneously.

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Shape Sensitivity Analysis and Design Optimization of Linear, Thermoelastic Solids

Cited by 10 publications

References 11 publications

Optimal airfoil shapes for low Reynolds number flows

Optimal airfoil shapes for low Reynolds number flows

Shape Optimization of Thermoelastic Fields for Mean Compliance Minimization

A stabilized finite element method for shape optimization in low Reynolds number flows

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