Formulation and Multigrid Solution of the Discrete Adjoint Problem on Unstructured Meshes

Mavriplis, Dimitri J.

doi:10.1007/3-540-31801-1_96

Cited by 2 publications

(3 citation statements)

References 10 publications

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“…40 Alternatively, some or all entries of the flow-Jacobian can be recomputed when forming the matrix-vector product, thereby reducing memory usage but increasing CPU time, see for example Giles et al, 15 Nielsen et al, 16 and Mavriplis. 17 To adopt the explicit flow solution method outlined in Sec. III for the solution of the adjoint and flowsensitivity equations, we implement a face-based algorithm to efficiently recompute the flow-Jacobian entries at each evaluation of the residual vector.…”

Section: B Numerical Implementationmentioning

confidence: 99%

“…The adjoint method, as well as the closely related flow-sensitivity (or direct) method, has been extensively analyzed and validated for the Euler and Navier-Stokes equations discretized on structured [11][12][13][14] and unstructured meshes. [15][16][17] When the adjoint equation is derived from the discrete form of the flow equations, which is the approach used in this work, the accuracy of the adjoint solution primarily depends on the accuracy of the linearization of the flow equations. The most common approach is to derive the linearization by hand.…”

Section: Introductionmentioning

confidence: 99%

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Adjoint Formulation for an Embedded-Boundary Cartesian Method

Nemec

Aftosmis

Murman

et al. 2005

43rd AIAA Aerospace Sciences Meeting and Exhibit

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A discrete-adjoint formulation is presented for the three-dimensional Euler equations discretized on a Cartesian mesh with embedded boundaries. The solution algorithm for the adjoint and flow-sensitivity equations leverages the Runge-Kutta time-marching scheme in conjunction with the parallel multigrid method of the flow solver. The matrix-vector products associated with the linearization of the flow equations are computed on-the-fly, thereby minimizing the memory requirements of the algorithm at a computational cost roughly equivalent to a flow solution. Three-dimensional test cases, including a wing-body geometry at transonic flow conditions and an entry vehicle at supersonic flow conditions, are presented. These cases verify the accuracy of the linearization and demonstrate the efficiency and robustness of the adjoint algorithm for complex-geometry problems.

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Section: B Numerical Implementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Adjoint Formulation for an Embedded-Boundary Cartesian Method

Nemec

Aftosmis

Murman

et al. 2005

43rd AIAA Aerospace Sciences Meeting and Exhibit

View full text Add to dashboard Cite

show abstract

“…Discrete adjoint method was derived from the discrete flow equations by Baysal et al [15,16]. Subsequently discrete adjoint method has been developed rapidly [17][18][19][20][21][22]. Discrete adjoint equations which come from discrete flow equations can provide a little more accurate gradients than continuous adjoint equations, but discrete adjoint equations are more difficult to be derived and implemented numerically.…”

Section: Introductionmentioning

confidence: 99%

Aerodynamic Optimization Based on Continuous Adjoint Method for a Flexible Wing

Xia

2016

International Journal of Aerospace Engineering

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Aerodynamic optimization based on continuous adjoint method for a flexible wing is developed using FORTRAN 90 in the present work. Aerostructural analysis is performed on the basis of high-fidelity models with Euler equations on the aerodynamic side and a linear quadrilateral shell element model on the structure side. This shell element can deal with both thin and thick shell problems with intersections, so this shell element is suitable for the wing structural model which consists of two spars, 20 ribs, and skin. The continuous adjoint formulations based on Euler equations and unstructured mesh are derived and used in the work. Sequential quadratic programming method is adopted to search for the optimal solution using the gradients from continuous adjoint method. The flow charts of rigid and flexible optimization are presented and compared. The objective is to minimize drag coefficient meanwhile maintaining lift coefficient for a rigid and flexible wing. A comparison between the results from aerostructural analysis of rigid optimization and flexible optimization is shown here to demonstrate that it is necessary to include the effect of aeroelasticity in the optimization design of a wing.

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Formulation and Multigrid Solution of the Discrete Adjoint Problem on Unstructured Meshes

Cited by 2 publications

References 10 publications

Adjoint Formulation for an Embedded-Boundary Cartesian Method

Adjoint Formulation for an Embedded-Boundary Cartesian Method

Aerodynamic Optimization Based on Continuous Adjoint Method for a Flexible Wing

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