Relaxation approaches to the optimal control of the Euler equations

Ngnotchouye, Jean Medard T.; Herty, Michaël; Steffensen, Sonja; Banda, Mapundi K.

doi:10.1590/s1807-03022011000200009

Cited by 10 publications

(4 citation statements)

References 36 publications

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“…As relaxation parameter we use = 10 −8 . More details on the specific numerical implementation can be found in [4]. In Figure 1 we display the results for the shock tube problem.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In [3] it is shown that under suitable assumptions, in the limit ε → 0, the macroscopic variables, that are derived by the solutions f i and the velocities ξ i , satisfy the original Euler equations. For the optimal control of the Euler equations, in [4] an associated adjoint system is derived via the usual Langrangian formalism. The adjoint equations in this case are given by…”

Section: Relaxation Methodsmentioning

confidence: 99%

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Optimal Control of the Euler Equations via Relaxation Approaches

Veelken¹,

Herty²,

Ngnotchouye³

et al. 2010

Proc Appl Math and Mech

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We discuss two linear relaxation approaches to the optimal control of nonlinear hyperbolic systems, in particular the control of Euler flows in gas dynamics. The first method is a relaxation system that is due to Jin and Xin [2], the second one is a Lattice-Boltzman approach [3], where we use one spatial dimension and five velocities (D1Q5 model). Both methods are incorporated in an adjoint based steepest-descent algorithm for the optimisation. Convincing numerical results are presented for both methods for an example with discontinuous solutions.We consider the optimal control of the one-dimensional Euler equations, which is of the formwhere u 0 is the control, u d is a given desired state and F : R n → R n is a nonlinear flux function, that in this case is given bywhere ρ is the density, m the momentum and E the energy of the considered gas. The velocity and the pressure of the gas can be derived by v = m/ρ and p = (γ−1)(E−1/2 ρ v), where γ denotes the specific heat ratio. Such kind of optimal control problems arise e.g. in gas dynamics and are difficult to solve mainly due to wave interactions that might occur in the solution of nonlinear systems of conservation laws. Relaxation MethodsRelaxation System: The first relaxation method that we are concerned with is the relaxation system that is proposed by Jin and Xin in [2]. The semi-linear approximation of the nonlinear hyperbolic equation ∂ t u + ∂ x F(u) = 0 with initial conditions u(x, 0) = u 0 is given bywhere ε > 0 is the relaxation rate and A = diag(a 1 , .

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“…As relaxation parameter we use = 10 −8 . More details on the specific numerical implementation can be found in [4]. In Figure 1 we display the results for the shock tube problem.…”

Section: Numerical Resultsmentioning

confidence: 99%

Section: Relaxation Methodsmentioning

confidence: 99%

Optimal Control of the Euler Equations via Relaxation Approaches

Veelken¹,

Herty²,

Ngnotchouye³

et al. 2010

Proc Appl Math and Mech

Self Cite

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“…Recently, developed the adjoint technique based on the standard LBM and applied it for flow optimization of the channel flow. Some researchers like Li et al (2018) or Ngnotchouye et al (2011) who have combined the adjoint approach and the LBM for the inverse problem, have developed it just for incompressible conditions using the Maxwell distribution function. There are several studies in developing the continuous-adjoint based optimization using the lattice Boltzmann method (Kreissl et al 2011, Pingen et al 2008, Tekitek et al 2006and Vergnault et al 2014.…”

Section: Introductionmentioning

confidence: 99%

Wave Drag Reduction of SC(2)0410 Airfoil using New Developed Inviscid Compressible Adjoint Method

2020

JAFM

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A supercritical airfoil is geometrically optimized using the new developed adjoint compressible lattice Boltzmann method. Minimizing the drag coefficient and eliminating the shock wave on the supercritical airfoil surface are considered as the cost function with constraint of fixed lift coefficient. The continuous adjoint method is applied to able designers to implement large number of design variables in actual optimization problems. The adjoint equation based on the specified cost function and constrains is successfully derived. Discretization of the governing equations is carried out using the finite volume approach and 3 rd order of the MUSCL scheme. The supercritical SC(2)0410 airfoil, which has a strong shock on the top surface at transonic cruise conditions, is numerically optimized using the inviscid developed algorithm to eliminate the shock and reduce the wave drag. To validate the obtained results and show viscosity effect on the results, the base airfoil and optimized one are experimentally tested in a transonic wind tunnel at the same conditions. Pressure distribution on the surface of both the base and optimal airfoil are extracted from the experimental tests and compared with those of numerical simulations. The results indicate that the developed approach can be properly used for supercritical airfoil shape optimization for elimination the shock and reduction the wave drag.

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“…However, direct applications of standard numerical schemes to the adjoint differential systems of the optimal control problem may lead to order reduction problems [19,33]. Besides classical applications to ODEs these problems gained interest recently in PDEs, in particular in the field of hyperbolic and kinetic equations [1,2,24,29].…”

Section: Introductionmentioning

confidence: 99%

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

Albi

Herty

Pareschi

2019

Applied Mathematics and Computation

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We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high-order accuracy. Subsequently we extend these results to semi-lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings.

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Relaxation approaches to the optimal control of the Euler equations

Cited by 10 publications

References 36 publications

Optimal Control of the Euler Equations via Relaxation Approaches

Optimal Control of the Euler Equations via Relaxation Approaches

Wave Drag Reduction of SC(2)0410 Airfoil using New Developed Inviscid Compressible Adjoint Method

Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

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