A two-level computational graph method for the adjoint of a finite volume based compressible unsteady flow solver

The adjoint method is a useful tool for finding gradients of design objectives with respect to system parameters for fluid dynamics simulations. But the utility of this method is hampered by the difficulty in writing an efficient implementation for the adjoint flow solver, especially one that scales to thousands of cores. This paper demonstrates a Python library, called adFVM, that can be used to construct an explicit unsteady flow solver and derive the corresponding discrete adjoint flow solver using automatic differentiation (AD). The library uses a two-level computational graph method for representing the structure of both solvers. The library translates this structure into a sequence of optimized kernels, significantly reducing its execution time and memory footprint. Kernels can be generated for heterogeneous architectures including distributed memory, shared memory and accelerator based systems. The library is used to write a finite volume based compressible flow solver. A wall clock time comparison between different flow solvers and adjoint flow solvers built using this library and state of the art graph based AD libraries is presented on a turbomachinery flow problem. Performance analysis of the flow solvers is carried out for CPUs and GPUs. Results of strong and weak scaling of the flow solver and its adjoint are demonstrated on subsonic flow in a periodic box.     

Toward optimally seeded airflow on hypersonic vehicles using control theory

Following the renewed interest in hypersonic flight and the significant advances made recently, it is now the time to start looking at ways to optimize hypersonic vehicle designs in an efficient manner. Since the medium, in a hypersonic flow, can be locally ionized, it is possible to use electromagnetic actuators that induce an acting force to optimally control the flow. The local injection of substances that have a considerably lower ionization temperature than air into the airflow - flow seeding - leads to stronger local ionization levels at relatively low hypersonic speeds, amplifying the magnetic effects for the same imposed magnetic field intensity. Because much has been devoted to the analysis of such problems but no formal design approach as been persued to date, the main motivation for this work is to provide an efficient design framework built around high-speed magnetohydrodynamics (MHD) prediction capabilities that can be used in hypersonic control applications using magnetic effects. In particular, the design framework should provide information that leads to an optimal airflow seeding strategy in conjunction with an imposed magnetic field. The proposed framework is based on control theory, which implies developing an adjoint solver aimed to efficiently provide sensitivity analysis capability in arbitrary complex hypersonics MHD flows. Automatic differentiation tools are selectively used to develop the discrete adjoint, which make for a much shorter implementation time and greatly reduce the probability of programming errors. A generic hypersonic vehicle is used to demonstrate the sensitivity analysis capability of the implemented MHD adjoint solver. The precision of the computed adjoint-based sensitivities is established and the performance of the adjoint solver is analyzed. A sample design problem is included using a gradient-based optimizer.     

A preference aggregation rule approach for structural optimization

In this paper, preference aggregation rules are used to define overall design evaluation measures in optimal design problems. A methodology for the efficient solution of the corresponding design optimization problems is presented. Each design criterion as well as the constraints imposed on the design variables and problem parameters are characterized by preference functions. The nondifferentiable nature of the optimization problems which arise in this formulation is coped with using a first-order algorithm combined with approximation concepts. High-quality approximations for the system response functions are constructed using the concepts of intermediate response quantities and intermediate variables. These approximations are used to replace the original problem by a sequence of approximate problems. Example problems are presented to study the performance of the proposed optimization technique as well as the methodology based on approximation concepts.     

Shape design optimization of an expansion step in a channel with moving boundaries for a viscous flow

A shape design optimization problem for viscous flows has been investigated in the present study. An analytical shape design sensitivity expression has been derived for a general integral functional by using the adjoint variable method and the material derivative concept of optimization. A channel flow problem with a backward facing step and adversely moving boundary wall is taken as an example. The shape profile of the expansion step, represented by a fourth-degree polynomial, is optimized in order to minimize the total viscous dissipation in the flow field. Numerical discretizations of the primary (flow) and adjoint problems are achieved by using the Galerkin FEM method. A balancing upwinding technique is also used in the equations. Numerical results are provided in various graphical forms at relatively low Reynolds numbers. It is concluded that the proposed general method of solution for shape design optimization problems is applicable to physical systems described by nonlinear equations.     

An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids

An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier-Stokes equations is presented. The method is constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system. The new approach is implemented within a three-dimensional unstructured-grid framework and results are presented for inviscid, laminar, and turbulent flows. Improvements to the baseline solution algorithm, such as line-implicit relaxation and a tight coupling of the turbulence model, are also presented. By storing nearest-neighbor terms in the residual computation, the dual scheme is computationally efficient, while requiring twice the memory of the flow solution. The current implementation allows for multiple right-hand side vectors, enabling simultaneous adjoint solutions for several cost functions or constraints with minimal additional storage requirements, while reducing the solution time compared to serial applications of the adjoint solver. The scheme is expected to have a broad impact on computational problems related to design optimization as well as error estimation and grid adaptation efforts.     

A hybrid algorithm for far-field noise minimization

A general unsteady adjoint formulation is applied to a hybrid acoustic prediction algorithm to provide an efficient far-field noise minimization algorithm. Two-dimensional unsteady Navier-Stokes (NS) computations for calculating the properties of acoustic sources are combined with the Ffowcs Williams and Hawkings (FW-H) wave propagation formulation to calculate the resulting far-field noise. Two different time-marching methods, namely an implicit multi-stage and an implicit multi-step method, are used for time discretization. The hybrid NS/FW-H solver is verified by comparison to an analytical solution and a Navier-Stokes solution. A discrete-adjoint Newton-Krylov algorithm is used to enable gradient-based shape optimization to minimize far-field noise computed using the hybrid solver. Objective functions considered include remote inverse shape designs for verification as well as the far-field pressure fluctuations for a blunt trailing edge airfoil in an unsteady turbulent flow environment. The examples presented demonstrate that the combination of a discrete-adjoint Newton-Krylov algorithm with a hybrid NS/FW-H far-field noise prediction method can be an efficient design tool for reducing aerodynamically generated noise.     

Aero-structural optimization using adjoint coupled post-optimality sensitivities

A new subspace optimization method for performing aero-structural design is introduced. The method relies on a semi-analytic adjoint approach to the sensitivity analysis that includes post-optimality sensitivity information from the structural optimization subproblem. The resulting coupled post-optimality sensitivity approach is used to guide a gradient-based optimization algorithm. The new approach simplifies the system-level problem, thereby reducing the number of calls to a potentially costly aerodynamics solver. The aero-structural optimization of an aircraft wing is performed using linear aerodynamic and structural analyses, and a performance comparison is made between the new approach and the conventional multidisciplinary feasible method. The new asymmetric suboptimization method is found to be the more efficient approach when it adequately reduces the number of system evaluations or when there is a large enough discrepancy between disciplinary solution times.     

Conceptual design of aeroelastic structures by topology optimization

A topology optimization methodology is presented for the conceptual design of aeroelastic structures accounting for the fluid–structure interaction. The geometrical layout of the internal structure, such as the layout of stiffeners in a wing, is optimized by material topology optimization. The topology of the wet surface, that is, the fluid–structure interface, is not varied. The key components of the proposed methodology are a Sequential Augmented Lagrangian method for solving the resulting large-scale parameter optimization problem, a staggered procedure for computing the steady-state solution of the underlying nonlinear aeroelastic analysis problem, and an analytical adjoint method for evaluating the coupled aeroelastic sensitivities. The fluid–structure interaction problem is modeled by a three-field formulation that couples the structural displacements, the flow field, and the motion of the fluid mesh. The structural response is simulated by a three-dimensional finite element method, and the aerodynamic loads are predicted by a three-dimensional finite volume discretization of a nonlinear Euler flow. The proposed methodology is illustrated by the conceptual design of wing structures. The optimization results show the significant influence of the design dependency of the loads on the optimal layout of flexible structures when compared with results that assume a constant aerodynamic load.     

A numerical approach for shape optimization of fluid flow domains

A numerical method for the shape optimization of fluid flow domains is presented and analyzed. The procedure is based on a flow solver, a mathematical optimization tool, and a technique for shape variation, which are combined into an integrated procedure. The flow solver relies on the discretization of the incompressible Navier–Stokes equations by means of the finite-volume method for block-structured, boundary-fitted grids with multi-grid acceleration. The optimization tool is an implementation of a trust region based derivative-free method. It is designed to minimize smooth functions whose evaluations are considered expensive and whose derivatives are not available or not desirable to approximate. The shape variation is obtained by deforming the computational grid employed by the flow solver. For this purpose, displacement fields scaled by the design variables are added to the initial grid. The displacement vectors are computed once before starting the optimization cycle by using a free-form deformation technique. Applications illustrating the functionality and the properties of the method are presented for some examples of engineering interest, such as the minimization of a pressure drop, the maximization of a lift force, and the optimization of a wall temperature.     

Iterative semi-continuous relaxation heuristics for the multiple-choice multidimensional knapsack problem

Recently several hybrid methods combining exact algorithms and heuristics have been proposed for solving hard combinatorial optimization problems. In this paper, we propose new iterative relaxation-based heuristics for the 0-1 Mixed Integer Programming problem (0-1 MIP), which generate a sequence of lower and upper bounds. The upper bounds are obtained from relaxations of the problem and refined iteratively by including pseudo-cuts in the problem. Lower bounds are obtained from the solving of restricted problems generated by exploiting information from relaxation and memory of the search process. We propose a new semi-continuous relaxation (SCR) that relaxes partially the integrality constraints to force the variables values close to 0 or 1. Several variants of the new iterative semi-continuous relaxation based heuristic can be designed by a given update procedure of multiplier of SCR. These heuristics are enhanced by using local search procedure to improve the feasible solution found and rounding procedure to restore infeasibility if possible. Finally we present computational results of the new methods to solve the multiple-choice multidimensional knapsack problem which is an NP-hard problem, even to find a feasible solution. The approach is evaluated on a set of problem instances from the literature, and compared to the results reached by both CPLEX solver and an efficient column generation-based algorithm. The results show that our algorithms converge rapidly to good lower bounds and visit new best-known solutions.     

Multi-material structural topology optimization with decision making of stiffness design criteria

A new methodology for making design decisions of structures using multi-material optimum topology information is presented. Multi-material analysis contributes significant applications to enhance the bearing capacity and performance of structures. A method that chooses an appropriate material combination satisfying design stiffness requirement economically is currently needed. An alternative method of making design-decision is to utilize a multi-material topology optimization (MMTO) approach. This study provides a new computational design optimization procedure as a guideline to find the optimal multi-material design by considering structure strain energy and material cost. The MMTO problem is analyzed using an alternative active-phase approach. The procedure consists of three design steps. First, steel grid configurations and composite with material properties are defined as a given structure for automatic design decision-making (DDM). And then design criteria of the steel composites structure is given to be limited strain energy by designers and engineers. Second, topology changes in the automatic distribution of multi-steel materials combination and volume control of each material during optimization procedures are achieved and at the same time, their converged minimal strain energy is produced for each material combination. And third, the strain energy and material cost which is computed based on the material ratio in the combinations are used as design decision parameters. A study in constructional steel composites to produce optimal and economical multi-material designs demonstrates the efficiency of the present DDM methodology.     

Hydrodynamic design using a derivative-free method

A derivative-free shape optimization tool for computational fluid dynamics (CFD) is developed in order to facilitate the implementation of complex flow solvers in the design procedure. A modified Rosenbrocks method is used, which needs neither gradient evaluations nor approximations. This approach yields a robust and flexible tool and gives the capability of performing optimizations involving complex configurations and phenomena. The flow solver implemented solves the Reynolds-averaged Navier–Stokes equations (RANSE) on unstructured grids, using near-wall, low-Reynolds-number turbulence models. Free surface effects are taken into account by a pseudosteady surface tracking method. A mesh deformation strategy based on both lineal and torsional springs analogies is used to update the mesh while maintaining the quality of the grid near the wall for two-dimensional problems. A free-form-deformation technique is used to manage the mesh and the shape perturbations for three-dimensional cases. Two hydrodynamic applications are presented, concerning first the design of a two-dimensional hydrofoil in relation with the free-surface elevation and then the three-dimensional optimization of a hull shape, at full scale.     

High-fidelity aerostructural optimization with integrated geometry parameterization and mesh movement

This paper extends an integrated geometry parameterization and mesh movement strategy for aerodynamic shape optimization to high-fidelity aerostructural optimization based on steady analysis. This approach provides an analytical geometry representation while enabling efficient mesh movement even for very large shape changes, thus facilitating efficient and robust aerostructural optimization. The geometry parameterization methodology uses B-spline surface patches to describe the undeflected design and flying shapes with a compact yet flexible set of parameters. The geometries represented are therefore independent of the mesh used for the flow analysis, which is an important advantage to this approach. The geometry parameterization is integrated with an efficient and robust grid movement algorithm which operates on a set of B-spline volumes that parameterize and control the flow grid. A simple technique is introduced to translate the shape changes described by the geometry parameterization to the internal structure. A three-field formulation of the discrete aerostructural residual is adopted, coupling the mesh movement equations with the discretized three-dimensional inviscid flow equations, as well as a linear structural analysis. Gradients needed for optimization are computed with a three-field coupled adjoint approach. Capabilities of the framework are demonstrated via a number of applications involving substantial geometric changes.     

Analysis and shape optimization in incompressible flows with unstructured grids

The aim of this study is to develop and validate numerical methods that perform shape optimization in incompressible flows using unstructured meshes. The three-dimensional Euler equations for compressible flow are modified using the idea of artificial compressibility and discretized on unstructured tetrahedral grids to provide estimates of pressure distributions for aerodynamic configurations. Convergence acceleration techniques like multigrid and residual averaging are used along with parallel computing platforms to enable these simulations to be performed in a few minutes. This computational frame-work is used to analyze sail geometries. The adjoint equations corresponding to the “incompressible” field equations are derived along with the functional form of gradients. The evaluation of the gradients is reduced to an integral around the boundary to circumvent hurdles posed by adjoint-based gradient evaluations on unstructured meshes. The reduced gradient evaluations provide major computational savings for unstructured grids and its accuracy and use for canonical and industrial problems is a major contribution of this study. The design process is driven by a steepest-descent algorithm with a fixed step-size. The feasibility of the design process is demonstrated for three inverse design problems, two canonical problems and one industrial problem.     

Structural optimal design of systems with imprecise properties: a possibilistic approach

The optimization problem of structural systems with imprecise properties on the basis of a possibilistic approach is considered. System imprecisions are defined by fuzzy numbers and characterized by membership functions. A methodology for the efficient solution of the optimization process is presented. A two-step method is used to include the imprecision within the optimization, where high quality approximations are used for the evaluation of structural responses. The approximations are constructed using concepts of intermediate response quantities and intermediate variables. The approach is basically an algebraic process which can be implemented very efficiently for the optimal design of general structural systems with imprecise parameters. The method provides more information to the designer than is available using conventional design tools. The effectiveness of the methodology and the interpretation of the results are illustrated by the solution of two example problems.     

Structural optimization of linear dynamical systems under stochastic excitation: a moving reliability database approach

A method to carry out structural synthesis of deterministic linear dynamical systems under stochastic excitation is introduced. The structural optimization problem is written as a nonlinear mathematical programming problem with reliability constraints. Probability that design conditions are satisfied within a given time period is used as a measure of system reliability. The solution of the original optimization problem is replaced by the solution of a sequence of approximate sub-optimization problems. An explicit approximation of the system reliability in terms of the design variables is constructed in each sub-optimization problem. The approximations are locally adjusted to a reliability database, which is obtained by an efficient importance sampling technique. Each approximate optimization problem is solved in an efficient manner due to the availability of the system reliability in explicit form. Numerical examples are presented to illustrate the performance and efficiency of the proposed methodology.     

On an extension of one-shot methods to incorporate additional constraints

For design optimization tasks, quite often a so-called one-shot approach is used. It augments the solution of the state equation with a suitable adjoint solver yielding approximate reduced derivatives that can be used in an optimization iteration to change the design. The coordination of these three iterative processes is well established when only the state equation is considered as equality constraint. However, numerous applications require also additional equality constraints. Therefore, we propose a modified augmented Lagrangian function, that defines a simultaneous change of all variables for this extended setting. It is shown that the augmented Lagrangian function proposed in this paper can be used in a gradient-based optimization approach to solve the original design task.     

A Modified Quasi-Newton Method for Optimization in Simulation

Optimization in Simulation is an important problem often encountered in system behavior investigation; however, the existing methods such as response surface methodology and stochastic approximation method are inefficient. This paper presents a modification of a quasi-Newton method, in which the parameters are determined from some numerical experiments. To demonstrate the validity of the devised method, two examples resembling the M/M/1 queueing problem are solved. The closeness of the converged solutions to the optimal solutions and a comparison with two stochastic approximation methods indicate that the modified quasi-Newton method as devised in this paper is a robust and efficient method for solving optimization problems in simulation.     

On multigrid-CG for efficient topology optimization

This article presents a computational approach that facilitates the efficient solution of 3-D structural topology optimization problems on a standard PC. Computing time associated with solving the nested analysis problem is reduced significantly in comparison to other existing approaches. The cost reduction is obtained by exploiting specific characteristics of a multigrid preconditioned conjugate gradients (MGCG) solver. In particular, the number of MGCG iterations is reduced by relating it to the geometric parameters of the problem. At the same time, accurate outcome of the optimization process is ensured by linking the required accuracy of the design sensitivities to the progress of optimization. The applicability of the proposed procedure is demonstrated on several 2-D and 3-D examples involving up to hundreds of thousands of degrees of freedom. Implemented in MATLAB, the MGCG-based program solves 3-D topology optimization problems in a matter of minutes. This paves the way for efficient implementations in computational environments that do not enjoy the benefits of high performance computing, such as applications on mobile devices and plug-ins for modeling software.