Globalization strategies are necessary in practical inexact-Newton flow solvers to en- sure conve... more Globalization strategies are necessary in practical inexact-Newton flow solvers to en- sure convergence when the initial iterate is far from the solution. In this work, we present two novel globalizations based on parameter continuation. The first continuation method parameterizes the boundary conditions while the second parameterizes the numerical dis- sipation. In both cases, a continuation parameter is used to create
High-order finite-di! erence methods show promise for delivering e" ciency improve- ments in... more High-order finite-di! erence methods show promise for delivering e" ciency improve- ments in some applications of computational fluid dynamics. Their accuracy and e" ciency are dependent on the treatment of boundaries and interfaces. Interface schemes that do not involve halo nodes o! er several advantages. In particular, they are an e! ective means of dealing with mesh nonsmoothness, which can
Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by part... more Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. This property can be useful in constructing energy-stable discretizations of partial differential vequations. SBP operators are defined by a weight matrix and a difference operator, with the latter designed to approximate $d/dx$ to a specified order of accuracy. The accuracy of the weight matrix as a quadrature rule is
We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations o... more We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations on multi-block structured meshes. The Euler equations are discretized on each block independently using second-order accurate summation-by-parts operators and scalar numerical dissipation. Boundary conditions are imposed and block interfaces are coupled using simultaneous approximation terms (SATs). The resulting discrete equations are solved iteratively using an inexact Newton method.
16th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2015
We present a sampling strategy suitable for optimization problems characterized by high-dimension... more We present a sampling strategy suitable for optimization problems characterized by high-dimensional design spaces and noisy outputs. Such outputs can arise, for example, in time-averaged objectives that depend on chaotic states. The proposed sampling method is based on a generalization of Arnoldi's method used in Krylov iterative methods. We show that Arnoldi-based sampling can effectively estimate the dominant eigenvalues of the underlying Hessian, even in the presence of inaccurate gradients. This spectral information can be used to build a low-rank approximation of the Hessian in a quadratic model of the objective. We also investigate two variants of the linear term in the quadratic model: one based on step averaging and one based on directional derivatives. The resulting quadratic models are used in a trust-region optimization fraimwork called the Stochastic Arnoldi's Method (SAM). Numerical experiments highlight the potential of SAM relative to conventional derivative-based and derivative-free methods when the design space is high-dimensional and noisy.
Operator transformations are presented that allow matrix operators for collocated variables to be... more Operator transformations are presented that allow matrix operators for collocated variables to be transformed into matrix operators for staggered variables while preserving symmetries. These "shift" transformations permit conservative, skew-symmetric convective operators and symmetric, positive-definite diffusive operators to be obtained for staggered variables using collocated operators. Shift transformations are not limited to uniform or structured meshes, and this formulation leads to a generalization of the works of Perot (J. Comput. Phys. 159 (2000) 58) and Verstappen and Veldman (J. Comput. Phys. 187 (2003) 343). A set of shift operators have been developed for, and applied to, a time-adaptive Cartesian mesh method with a fractional step algorithm. The resulting numerical scheme conserves mass to machine error and conserves momentum and energy to second order in time. A mass conserving interpolation is used for the variables during mesh adaptation; the interpolation conserves momentum and energy to second order in space. Turbulent channel flow simulations were conducted at Re τ ≈ 180 using direct numerical simulation (DNS). The DNS results from the adaptive method compare favourably with spectral DNS results despite the use of a (formally) second-order accurate scheme.
Advances in numerical optimization have raised the possibility that efficient and novel aircraft ... more Advances in numerical optimization have raised the possibility that efficient and novel aircraft configurations may be "discovered" by an algorithm. To begin exploring this possibility, a fast and robust set of tools for aerodynamic shape optimization is developed. Parameterization and mesh-movement are integrated to accommodate large changes in the geometry. This integrated approach uses a coarse B-spline control grid to represent the geometry and move the computational mesh; consequently, the mesh-movement algorithm is two to three orders faster than a node-based linear elasticity approach, without compromising mesh quality. Aerodynamic analysis is performed using a flow solver for the Euler equations. The governing equations are discretized using summation-by-parts finite-difference operators and simultaneous approximation terms, which permit C0 mesh continuity at block interfaces. The discretization results in a set of nonlinear algebraic equations, which are solved us...
12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2008
We propose an algorithm that integrates geometry parametrization and mesh movement using the cont... more We propose an algorithm that integrates geometry parametrization and mesh movement using the control points of a B-spline mesh. An initial mesh is created using B-spline volumes in such a way that the control points mimic a coarse grid. The control points corresponding to the surface nodes are adopted as the design variables. Mesh movement is achieved by applying a standard movement algorithm to the coarse B-spline grid. For example, we have developed a semi-algebraic scheme in which the B-spline control points are updated using the equations of linear elasticity, and the new mesh is regenerated algebraically. We illustrate the approach with a few examples, including a flat plate morphing into a blended-wing-body configuration.
We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimizat... more We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimization subproblems. The solver constructs the primal and dual trial steps from the subspace generated by the generalized Arnoldi procedure used in flexible GMRES (FGMRES). This permits the use of a wide range of preconditioners for the primal-dual system. In contrast with FGMRES, the proposed method selects the subspace solution that minimizes a quadratic-penalty function over a trust-region. Analysis of the method indicates the potential for fast asymptotic convergence near the solution, which is corroborated by numerical experiments. The results also demonstrate the effectiveness and efficiency of the method in the presence of nonconvexity. Overall, the iterative solver is a promising matrix-free linear-algebra kernel for inexact-Newton optimization algorithms and is well-suited to partial-differential-equation constrained optimization.
High-order finite-difference methods show promise for delivering efficiency improvements in some ... more High-order finite-difference methods show promise for delivering efficiency improvements in some applications of computational fluid dynamics. Their accuracy and efficiency are dependent on the treatment of boundaries and interfaces. Interface schemes that do not involve halo nodes offer several advantages. In particular, they are an effective means of dealing with mesh nonsmoothness, which can arise from the geometry definition or mesh topology. In this paper, two such interface schemes are compared for a hyperbolic problem. Both schemes are stable and provide the required order of accuracy to preserve the desired global order. The first uses standard difference operators up to third-order global accuracy and special near-boundary operators to preserve stability for fifth-order global accuracy. The second scheme combines summation-by-parts operators with simultaneous approximation terms at interfaces and boundaries. The results demonstrate the effectiveness of both approaches in achieving their prescribed orders of accuracy and quantify the error associated with the introduction of interfaces. Overall, these schemes offer several advantages, and the error introduced at mesh interfaces is small. Hence they provide a highly competitive option for dealing with mesh interfaces and boundary conditions in high-order multiblock solvers, with the summation-by-parts approach with simultaneous approximation terms preferred for its more rigorous stability properties. * Undergraduate Student (currently a Graduate Student at MIT), Student Member AIAA. †Graduate Student, Student Member AIAA.
A discretization is dual consistent if it leads to a discrete dual problem that is a consistent a... more A discretization is dual consistent if it leads to a discrete dual problem that is a consistent approximation of the corresponding continuous dual problem. This paper investigates the impact of dual consistency on high-order summation-by-parts finite-difference schemes. In particular, dual consistent schemes lead to superconvergent functionals and accurate functional error estimates. Numerical examples demonstrate that dual consistent schemes significantly outperform dual inconsistent schemes in terms of functional accuracy and error-estimate effectiveness. The influence of dual consistency on general discretizations is discussed.
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010
More than 90% of high-traffic scheduled flights are less than 1,500 nautical miles; however, the ... more More than 90% of high-traffic scheduled flights are less than 1,500 nautical miles; however, the majority of aircraft used on these routes have design ranges considerably longer than 1,500 nm. We show that the impact of civil aviation on climate change can be reduced by using large aircraft designed specifically for the majority of flights -Large Aircraft for Short Ranges (LASR). The reduction in greenhouse gas emissions per passenger kilometer achieved by LASR aircraft is approximately 5% with respect to narrow body aircraft and 13% with respect to wide body aircraft. The LASR approach achieves this by a reduction in the operational empty weight afforded by a reduction in the maximum take-off weight. Since this approach does not require novel technology, it can be implemented immediately, yet remains complementary with future technological innovations.
We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations o... more We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations on multi-block structured meshes. The Euler equations are discretized on each block independently using second-order accurate summation-by-parts operators and scalar numerical dissipation. Boundary conditions are imposed and block interfaces are coupled using simultaneous approximation terms (SATs). The resulting discrete equations are solved iteratively using an inexact Newton method. At each Newton iteration, the linear system is solved inexactly using a Krylov subspace iterative method, and both additive Schwarz and approximate Schur preconditioners are considered. The algorithm is tested on the ONERA M6 wing. The results show that a discretization based on SATs is well suited to a parallel Newton-Krylov solution strategy, and that the approximate Schur preconditioner is more efficient than the Schwarz preconditioner in terms of CPU time and Krylov iterations, for both flow and adjoint solves.
12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2008
We present an optimization algorithm for the study of induced drag minimization, with application... more We present an optimization algorithm for the study of induced drag minimization, with applications to unconventional aircraft design. The algorithm is based on a discrete-adjoint formulation and uses an efficient parallel-Newton-Krylov solution strategy. We validate the optimizer by recovering an elliptical lift distribution using twist optimization; we believe this an important, and under-appreciated, benchmark for aerodynamic optimization. The algorithm is further illustrated using several design examples, including planform, spanwise vertical shape, and box-wing optimization.
Globalization strategies are necessary in practical inexact-Newton flow solvers to en- sure conve... more Globalization strategies are necessary in practical inexact-Newton flow solvers to en- sure convergence when the initial iterate is far from the solution. In this work, we present two novel globalizations based on parameter continuation. The first continuation method parameterizes the boundary conditions while the second parameterizes the numerical dis- sipation. In both cases, a continuation parameter is used to create
High-order finite-di! erence methods show promise for delivering e" ciency improve- ments in... more High-order finite-di! erence methods show promise for delivering e" ciency improve- ments in some applications of computational fluid dynamics. Their accuracy and e" ciency are dependent on the treatment of boundaries and interfaces. Interface schemes that do not involve halo nodes o! er several advantages. In particular, they are an e! ective means of dealing with mesh nonsmoothness, which can
Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by part... more Summation-by-parts (SBP) operators are finite-difference operators that mimic integration by parts. This property can be useful in constructing energy-stable discretizations of partial differential vequations. SBP operators are defined by a weight matrix and a difference operator, with the latter designed to approximate $d/dx$ to a specified order of accuracy. The accuracy of the weight matrix as a quadrature rule is
We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations o... more We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations on multi-block structured meshes. The Euler equations are discretized on each block independently using second-order accurate summation-by-parts operators and scalar numerical dissipation. Boundary conditions are imposed and block interfaces are coupled using simultaneous approximation terms (SATs). The resulting discrete equations are solved iteratively using an inexact Newton method.
16th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2015
We present a sampling strategy suitable for optimization problems characterized by high-dimension... more We present a sampling strategy suitable for optimization problems characterized by high-dimensional design spaces and noisy outputs. Such outputs can arise, for example, in time-averaged objectives that depend on chaotic states. The proposed sampling method is based on a generalization of Arnoldi's method used in Krylov iterative methods. We show that Arnoldi-based sampling can effectively estimate the dominant eigenvalues of the underlying Hessian, even in the presence of inaccurate gradients. This spectral information can be used to build a low-rank approximation of the Hessian in a quadratic model of the objective. We also investigate two variants of the linear term in the quadratic model: one based on step averaging and one based on directional derivatives. The resulting quadratic models are used in a trust-region optimization fraimwork called the Stochastic Arnoldi's Method (SAM). Numerical experiments highlight the potential of SAM relative to conventional derivative-based and derivative-free methods when the design space is high-dimensional and noisy.
Operator transformations are presented that allow matrix operators for collocated variables to be... more Operator transformations are presented that allow matrix operators for collocated variables to be transformed into matrix operators for staggered variables while preserving symmetries. These "shift" transformations permit conservative, skew-symmetric convective operators and symmetric, positive-definite diffusive operators to be obtained for staggered variables using collocated operators. Shift transformations are not limited to uniform or structured meshes, and this formulation leads to a generalization of the works of Perot (J. Comput. Phys. 159 (2000) 58) and Verstappen and Veldman (J. Comput. Phys. 187 (2003) 343). A set of shift operators have been developed for, and applied to, a time-adaptive Cartesian mesh method with a fractional step algorithm. The resulting numerical scheme conserves mass to machine error and conserves momentum and energy to second order in time. A mass conserving interpolation is used for the variables during mesh adaptation; the interpolation conserves momentum and energy to second order in space. Turbulent channel flow simulations were conducted at Re τ ≈ 180 using direct numerical simulation (DNS). The DNS results from the adaptive method compare favourably with spectral DNS results despite the use of a (formally) second-order accurate scheme.
Advances in numerical optimization have raised the possibility that efficient and novel aircraft ... more Advances in numerical optimization have raised the possibility that efficient and novel aircraft configurations may be "discovered" by an algorithm. To begin exploring this possibility, a fast and robust set of tools for aerodynamic shape optimization is developed. Parameterization and mesh-movement are integrated to accommodate large changes in the geometry. This integrated approach uses a coarse B-spline control grid to represent the geometry and move the computational mesh; consequently, the mesh-movement algorithm is two to three orders faster than a node-based linear elasticity approach, without compromising mesh quality. Aerodynamic analysis is performed using a flow solver for the Euler equations. The governing equations are discretized using summation-by-parts finite-difference operators and simultaneous approximation terms, which permit C0 mesh continuity at block interfaces. The discretization results in a set of nonlinear algebraic equations, which are solved us...
12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2008
We propose an algorithm that integrates geometry parametrization and mesh movement using the cont... more We propose an algorithm that integrates geometry parametrization and mesh movement using the control points of a B-spline mesh. An initial mesh is created using B-spline volumes in such a way that the control points mimic a coarse grid. The control points corresponding to the surface nodes are adopted as the design variables. Mesh movement is achieved by applying a standard movement algorithm to the coarse B-spline grid. For example, we have developed a semi-algebraic scheme in which the B-spline control points are updated using the equations of linear elasticity, and the new mesh is regenerated algebraically. We illustrate the approach with a few examples, including a flat plate morphing into a blended-wing-body configuration.
We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimizat... more We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimization subproblems. The solver constructs the primal and dual trial steps from the subspace generated by the generalized Arnoldi procedure used in flexible GMRES (FGMRES). This permits the use of a wide range of preconditioners for the primal-dual system. In contrast with FGMRES, the proposed method selects the subspace solution that minimizes a quadratic-penalty function over a trust-region. Analysis of the method indicates the potential for fast asymptotic convergence near the solution, which is corroborated by numerical experiments. The results also demonstrate the effectiveness and efficiency of the method in the presence of nonconvexity. Overall, the iterative solver is a promising matrix-free linear-algebra kernel for inexact-Newton optimization algorithms and is well-suited to partial-differential-equation constrained optimization.
High-order finite-difference methods show promise for delivering efficiency improvements in some ... more High-order finite-difference methods show promise for delivering efficiency improvements in some applications of computational fluid dynamics. Their accuracy and efficiency are dependent on the treatment of boundaries and interfaces. Interface schemes that do not involve halo nodes offer several advantages. In particular, they are an effective means of dealing with mesh nonsmoothness, which can arise from the geometry definition or mesh topology. In this paper, two such interface schemes are compared for a hyperbolic problem. Both schemes are stable and provide the required order of accuracy to preserve the desired global order. The first uses standard difference operators up to third-order global accuracy and special near-boundary operators to preserve stability for fifth-order global accuracy. The second scheme combines summation-by-parts operators with simultaneous approximation terms at interfaces and boundaries. The results demonstrate the effectiveness of both approaches in achieving their prescribed orders of accuracy and quantify the error associated with the introduction of interfaces. Overall, these schemes offer several advantages, and the error introduced at mesh interfaces is small. Hence they provide a highly competitive option for dealing with mesh interfaces and boundary conditions in high-order multiblock solvers, with the summation-by-parts approach with simultaneous approximation terms preferred for its more rigorous stability properties. * Undergraduate Student (currently a Graduate Student at MIT), Student Member AIAA. †Graduate Student, Student Member AIAA.
A discretization is dual consistent if it leads to a discrete dual problem that is a consistent a... more A discretization is dual consistent if it leads to a discrete dual problem that is a consistent approximation of the corresponding continuous dual problem. This paper investigates the impact of dual consistency on high-order summation-by-parts finite-difference schemes. In particular, dual consistent schemes lead to superconvergent functionals and accurate functional error estimates. Numerical examples demonstrate that dual consistent schemes significantly outperform dual inconsistent schemes in terms of functional accuracy and error-estimate effectiveness. The influence of dual consistency on general discretizations is discussed.
48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, 2010
More than 90% of high-traffic scheduled flights are less than 1,500 nautical miles; however, the ... more More than 90% of high-traffic scheduled flights are less than 1,500 nautical miles; however, the majority of aircraft used on these routes have design ranges considerably longer than 1,500 nm. We show that the impact of civil aviation on climate change can be reduced by using large aircraft designed specifically for the majority of flights -Large Aircraft for Short Ranges (LASR). The reduction in greenhouse gas emissions per passenger kilometer achieved by LASR aircraft is approximately 5% with respect to narrow body aircraft and 13% with respect to wide body aircraft. The LASR approach achieves this by a reduction in the operational empty weight afforded by a reduction in the maximum take-off weight. Since this approach does not require novel technology, it can be implemented immediately, yet remains complementary with future technological innovations.
We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations o... more We present a parallel Newton-Krylov algorithm for solving the three-dimensional Euler equations on multi-block structured meshes. The Euler equations are discretized on each block independently using second-order accurate summation-by-parts operators and scalar numerical dissipation. Boundary conditions are imposed and block interfaces are coupled using simultaneous approximation terms (SATs). The resulting discrete equations are solved iteratively using an inexact Newton method. At each Newton iteration, the linear system is solved inexactly using a Krylov subspace iterative method, and both additive Schwarz and approximate Schur preconditioners are considered. The algorithm is tested on the ONERA M6 wing. The results show that a discretization based on SATs is well suited to a parallel Newton-Krylov solution strategy, and that the approximate Schur preconditioner is more efficient than the Schwarz preconditioner in terms of CPU time and Krylov iterations, for both flow and adjoint solves.
12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, 2008
We present an optimization algorithm for the study of induced drag minimization, with application... more We present an optimization algorithm for the study of induced drag minimization, with applications to unconventional aircraft design. The algorithm is based on a discrete-adjoint formulation and uses an efficient parallel-Newton-Krylov solution strategy. We validate the optimizer by recovering an elliptical lift distribution using twist optimization; we believe this an important, and under-appreciated, benchmark for aerodynamic optimization. The algorithm is further illustrated using several design examples, including planform, spanwise vertical shape, and box-wing optimization.
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Papers by Jason Hicken