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Get to Know Us. English Choose a language for shopping. Audible Download Audio Books. Alexa Actionable Analytics for the Web. The correction strategy as well as the algorithm presented below can be found in [7,9]. The one-shot algorithm for solving this problem reads as follows:. As it is well-known, multigrid methods are iterative methods for solving system of algebraic equations, which results, for example, after discretization of PDEs.
In these methods, iterative methods such as Jacobi, GaussSeidel etc. Pioneering works, theories and applications of multigrid methods can be found, among others, in . In our applications, we use similar strategy. The basic difference in our implementations is that we use multigrid in the context of simultaneous pseudo-time stepping.
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In the multigrid algorithm we solve different optimization subproblems on different grid levels. The coarse grid solution, which is achieved in much less computational effort, is used to accelerate the convergence of the optimization problem in fine grid. The solution methodology is based on simultaneous pseudo-time stepping as mentioned in the previous section.
The ideas are demonstrated through practical aerodynamic shape optimization applications with Euler equations in 2D. In the following algorithm, n denotes the current mesh resolution and N the next coarser mesh resolution; denotes the prolongation operator and denotes the restriction operator. The multigrid algorithm reads as . The algorithm defines the V-cycle template of the multigrid algorithm.
The coarse grid subproblem contains objective function and the state constraints which are corrected using the in-exact reduced gradients. The steps based on coarse-grid subproblem leads to faster improvement of the optimal solution of the fine-grid problem maintaining the feasibility. Multigrid computations start on the finest level.
For prolongation we use linear interpolation and for restriction we use simple injection. Since problems 7 and 10 are of similar structure, all steps of Algorithm 1 can be carried out at respective discretization levels. Only for problem 10 , step 6 of Algorithm 1 is carried out after the prolongation of the update vector to finest level.
This has advantage that all the modules of the codes, developed in our earlier works, can be used in different discretization levels with minor modifications. In the present paper, we have implemented this method for the shape design example using Euler equations.
The details of governing equations, discretization, geometry parameterization, gradient computation and grid perturbation strategy can be found in [3,7]. We present here the results of a test case of RAE airfoil for drag reduction with constant lift together with geometric constraint of constant thickness. The flow conditions are given at transonic Mach number 0. Further results obtained using this method can be found in .
On these grid levels the preconditioned pseudo-stationary equations, resulting from the necessary optimality conditions corresponding to the optimization subproblems of Section 3, are solved using Algorithm 1. The initial values of the iterations i.
After the convergence of the optimization problem, another time-steps are carried out in L1 for the state equation to get more accurate values of the surface pressure and force coefficients. The number of design parameters on three levels are 81 L1 , 41 L2 and 21 L3 respectively. In step ii of Algorithm 2, the reduced gradients are the sum of reduced gradients at levels L1 and L2. On the finest level, 5 iterations and on the next two coarser levels 6 and 2 iterations, respectively, are carried out.
In the prolongation step from L3 to L2, 2 iterations are carried out.
In the next restriction from L2 to L3, 2 iterations are carried out. In the next prolongation step from L3 to L2, 6 iteration as are carried out. The convergence of the optimization requires 2 W-cycles and in total 95 optimization iterations. For the same problem, computation on single grid using Algorithm 1 requires iterations see in . The drag reduction is about These values are same as those of single grid computations.
The optimization convergence history is presented in Figure 1 , where the change in darg top and in lift bottom values during the optimization iterations appear. The convergence history of state and costate iterations are presented in Figure 2 at different grid levels. Comparison of cam ber lines, airfoils and surface pressure distributions are presented in Figure 3 , where one can see in the surface pressure distributions bottom that the strong shock before optimization has been disappeared for the optimal profile.
The number of optimization iterations depend on parameters chosen in the reduced Hessian updates. By fine tuning those parameters, same convergence criteria could. Figure 1. Convergence history of the optimization iterations. Figure 2. Convergence history of state and costate residuals on level-1 a , level-2 b and level-3 c. However, it is to note that the final converged solution remains the same. This is another proof of numerical stability of the method. Also in any case, the total effort is about 5 times that of the forward simulation runs, since one sufficiently converged forward solution needs about time-steps and total effort in the finest level is less than four forward simulation runs.
Adding up the time-steps required in coarser and coarsest level, we come up with this number. One-shot multigrid strategy is explained and used for aerodynamic shape optimization. The preconditioned pseudo-stationary state, costate and design equations are integrated simultaneously in time at different discretiza-. Figure 3. Comparison of Camberlines, airfoils and surface pressure distributions.
Coarse grid solution, which is less expensive to compute, is used to accelerate the convergence of the optimization problem in fine grid. Multigrid strategy is quite effective when far away from the solution.
Real-Time PDE-Constrained Optimization (Computational Science and Engineering)
Hazra and V. Hazra, V.
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