902 research outputs found
Development of the Discrete Adjoint for a Three-Dimensional Unstructured Euler Solver
The discrete adjoint of a reconstruction-based unstructured finite volume formulation for the Euler equations is derived and implemented. The matrix-vector products required to solve the adjoint equations are computed on-the-fly by means of an efficient two-pass assembly. The adjoint equations are solved with the same solution scheme adopted for the flow equations. The scheme is modified to efficiently account for the simultaneous solution of several adjoint equations. The implementation is demonstrated on wing and wing–body configurations
Towards Reduced-order Model Accelerated Optimization for Aerodynamic Design
The adoption of mathematically formal simulation-based optimization approaches within aerodynamic design depends upon a delicate balance of affordability and accessibility. Techniques are needed to accelerate the simulation-based optimization process, but they must remain approachable enough for the implementation time to not eliminate the cost savings or act as a barrier to adoption.
This dissertation introduces a reduced-order model technique for accelerating fixed-point iterative solvers (e.g. such as those employed to solve primal equations, sensitivity equations, design equations, and their combination). The reduced-order model-based acceleration technique collects snapshots of early iteration (pre-convergent) solutions and residuals and then uses them to project to significantly more accurate solutions, i.e. smaller residual. The technique can be combined with other convergence schemes like multigrid and adaptive timestepping. The technique is generalizable and in this work is demonstrated to accelerate steady and unsteady flow solutions; continuous and discrete adjoint sensitivity solutions; and one-shot design optimization solutions. This final application, reduced-order model accelerated one-shot optimization approach, in particular represents a step towards more efficient aerodynamic design optimization.
Through this series of applications, different basis vectors were considered and best practices for snapshot collection procedures were outlined. The major outcome of this dissertation is the development and demonstration of this reduced-order model acceleration technique. This work includes the first application of the reduced-order model-based acceleration method to an explicit one-shot iterative optimization process
Flow simulation and shape optimization for aircraft design
AbstractWithin the framework of the German aerospace research program, the CFD project MEGADESIGN was initiated. The main goal of the project is the development of efficient numerical methods for shape design and optimization. In order to meet the requirements of industrial implementations a co-operative effort has been set up which involves the German aircraft industry, the DLR, several universities and some small enterprises specialized in numerical optimization. This paper outlines the planned activities within MEGADESIGN, the status at the beginning of the project and it presents some early results achieved in the project
Simultaneous single-step one-shot optimization with unsteady PDEs
The single-step one-shot method has proven to be very efficient for PDE-constrained optimization where the partial differential equation (PDE) is solved by an iterative fixed point solver. In this approach, the simulation and optimization tasks are performed simultaneously in a single iteration. If the PDE is unsteady, finding an appropriate fixed point iteration is non-trivial. In this paper, we provide a framework that makes the single-step one-shot method applicable for unsteady PDEs that are solved by classical time-marching schemes. The one-shot method is applied to an optimal control problem with unsteady incompressible Navier-Stokes equations that are solved by an industry standard simulation code. With the Van-der-Pol oscillator as a generic model problem, the modified simulation scheme is further improved using adaptive time scales. Finally, numerical results for the advection-diffusion equation are presented. Keywords: Simultaneous optimization; One-shot method; PDE-constrained optimization; Unsteady PDE; Adaptive time scal
Deformable Overset Grid for Multibody Unsteady Flow Simulation
A deformable overset grid method is proposed to simulate the unsteady aerodynamic problems with multiple flexible moving bodies. This method uses an unstructured overset grid coupled with local mesh deformation to achieve both robustness and efficiency. The overset grid hierarchically organizes the subgrids into clusters and layers, allowing for overlapping/embedding of different type meshes, in which the mesh quality and resolution can be independently controlled. At each time step, mesh deformation is locally applied to the subgrids associated with deforming bodies by an improved Delaunay graph mapping method that uses a very coarse Delaunay mesh as the background graph. The graph is moved and deformed by the spring analogy method according to the specified motion, and then the computational meshes are relocated by a simple one-to-one mapping. An efficient implicit hole-cutting and intergrid boundary definition procedure is implemented fully automatically for both cell-centered and cell-vertex schemes based on the wall distance and an alternative digital tree data search algorithm. This method is successfully applied to several complex multibody unsteady aerodynamic simulations, and the results demonstrate the robustness and efficiency of the proposed method for complex unsteady flow problems, particularly for those involving simultaneous large relative motion and self-deformation
Adjoint-based aerodynamic shape optimization on unstructured meshes
In this paper, the exact discrete adjoint of an unstructured finite-volume formulation of the Euler equations in two dimensions is derived and implemented. The adjoint equations are solved with the same implicit scheme as used for the flow equations. The scheme is modified to efficiently account for multiple functionals simultaneously. An optimization framework, which couples an analytical shape parameterization to the flow/adjoint solver and to algorithms for constrained optimization, is tested on airfoil design cases involving transonic as well as supersonic flows. The effect of some approximations in the discrete adjoint, which aim at reducing the complexity of the implementation, is shown in terms of optimization results rather than only in terms of gradient accuracy. The shape-optimization method appears to be very efficient and robust
Combining Parameterizations, Sobolev Methods and Shape Hessian Approximations for Aerodynamic Design Optimization
Aerodynamic design optimization, considered in this thesis, is a large and complex area spanning different disciplines from mathematics to engineering. To perform optimizations on industrially relevant test cases, various algorithms and techniques have been proposed throughout the literature, including the Sobolev smoothing of gradients. This thesis combines the Sobolev methodology for PDE constrained flow problems with the parameterization of the computational grid and interprets the resulting matrix as an approximation of the reduced shape Hessian.
Traditionally, Sobolev gradient methods help prevent a loss of regularity and reduce high-frequency noise in the derivative calculation. Such a reinterpretation of the gradient in a different Hilbert space can be seen as a shape Hessian approximation. In the past, such approaches have been formulated in a non-parametric setting, while industrially relevant applications usually have a parameterized setting. In this thesis, the presence of a design parameterization for the shape description is explicitly considered. This research aims to demonstrate how a combination of Sobolev methods and parameterization can be done successfully, using a novel mathematical result based on the generalized Faà di Bruno formula. Such a formulation can yield benefits even if a smooth parameterization is already used.
The results obtained allow for the formulation of an efficient and flexible optimization strategy, which can incorporate the Sobolev smoothing procedure for test cases where a parameterization describes the shape, e.g., a CAD model, and where additional constraints on the geometry and the flow are to be considered. Furthermore, the algorithm is also extended to One Shot optimization methods. One Shot algorithms are a tool for simultaneous analysis and design when dealing with inexact flow and adjoint solutions in a PDE constrained optimization. The proposed parameterized Sobolev smoothing approach is especially beneficial in such a setting to ensure a fast and robust convergence towards an optimal design.
Key features of the implementation of the algorithms developed herein are pointed out, including the construction of the Laplace-Beltrami operator via finite elements and an efficient evaluation of the parameterization Jacobian using algorithmic differentiation. The newly derived algorithms are applied to relevant test cases featuring drag minimization problems, particularly for three-dimensional flows with turbulent RANS equations. These problems include additional constraints on the flow, e.g., constant lift, and the geometry, e.g., minimal thickness. The Sobolev smoothing combined with the parameterization is applied in classical and One Shot optimization settings and is compared to other traditional optimization algorithms. The numerical results show a performance improvement in runtime for the new combined algorithm over a classical Quasi-Newton scheme
Aerodynamic shape optimization by means of sequential linear programming techniques
A Sequential Linear Programming technique, known as the method of centers, is the driver of an aerodynamic shape optimization framework on unstructured meshes. The first order information required by this technique, functional values and their gradients, are computed by a median-dual flow/adjoint solver which is coupled to an analytical shape parameterization. Functional and geometric constraints are easily handled by the algorithm which appears to be very effective in obtaining efficiently near-optimal designs. Shape optimization results are presented for transonic as well as supersonic flows involving appreciable shape deformations
A coupled discrete adjoint method for optimal design with dynamic non-linear fluid structure interactions
Incorporating high-fidelity analysis methods in multidisciplinary design optimization necessitates efficient sensitivity evaluation, which is particularly important for time-accurate problems. This thesis presents a new discrete adjoint formulation suitable for fully coupled, non-linear, dynamic FSI problems. The solution includes time-dependent adjoint variables that arise from grid motion and chosen time integration methods for both the fluid and structural domains. Implemented as a generic multizone discrete adjoint solver for time-accurate analysis in the open-source multiphysics solver SU2, this provides a flexible framework for a wide range of applications. Design optimization of aerodynamic structures need accurate characterization of the coupled
fluid-structure interactions (FSI). Incorporating high-fidelity analysis methods in the multidisciplinary design optimization (MDO) necessitates efficient sensitivity evaluation, which is
particularly important for time-accurate problems. Adjoint methods are well established for
sensitivity analysis when large number of design variables are needed. The use of discrete adjoint method through algorithmic differentiation enables the evaluation of sensitivities using
an approximation of the Jacobian of the coupled problem, thus enabling this approach to be
applied for multidisciplinary analysis.
This thesis presents a new discrete adjoint formulation suitable for fully coupled, non-linear,
dynamic FSI problems. A partitioned approach is considered with finite volume for the fluid
and finite elements for the solid domains. The solution includes the time-dependent adjoint
variables that arise from the grid motion and chosen time integration methods for both the fluid
and structural domains. Implemented as a generic multizone discrete adjoint solver for timeaccurate analysis in the open-source multiphysics solver SU2, this provides a flexible framework
for a wide range of applications. The partitioned FSI solver approach has been leveraged to
extend the dynamic FSI capabilities to low speed flows through the introduction of a densitybased unsteady incompressible flow solver.
The developed methodology and implementation are demonstrated using a range of numerical
test cases. Optimal design for steady, coupled FSI problems are firstly presented before moving
to the building blocks of dynamic coupled problems using single domain analysis, for both
structural and fluid domains in turn. The new unsteady incompressible fluid solver, for both
the primal and adjoint analysis, are verified against a range of well-known benchmark test
cases, including problems with grid motion. Finally, applications of coupled dynamic problems
are presented to verify both the unsteady incompressible solver for FSI as well as the successful
verification of the discrete adjoint sensitivities for the transient response of a transonic compliant
airfoil for a variety of both aerodynamic and structural objective functions.Open Acces
- …