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Discontinuous Galerkin On Graphics Processing Units

Published: November 24, 2020

Discontinuous Galerkin (DG) is a popular class of numerical methods for solving partial differential equations (PDEs). Like any other finite element method, the solution inside each mesh element is approximated by a set of basis functions in DG. However, as indicated by the name, no continuity restrictions are enforced at element interfaces. The coupling between adjacent elements only comes in with uniquely defined inter-element numerical fluxes. The discontinuity nature might be counterintuitive at first glance, while it buys DG many advantages over traditional numerical methods.

Some Interesting Linear Algebra Problems

Published: November 23, 2020

In this blog, I collected some interesting (I thought) theorems, conclusions, and problems in linear algebra. It’s mainly for my own reference and for fun! I’ll update them and related proofs frequently ;).

Hello World

Published: June 12, 2019

Hello world! This is a test blog! My personal web just came online, I’ll keep updating it ;)

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Discretization error control for constrained aerodynamic shape optimization

Published in , 2009

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

High-Reynolds Number Transitional Flow Simulation via Parabolized Stability Equations with an Adaptive RANS Solver

Published in , 2009

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

A method to Study Airfoil Dynamic Stall Based on Transition Point

Published in , 2015

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

Airfoil shape optimization using output-based adapted meshes

Published in , 2017

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

High-Reynolds Number Transitional Flow Prediction using a Coupled Discontinuous-Galerkin RANS PSE Framework

Published in , 2019

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

Output-Based Mesh Adaptation for Variable-Fidelity Multipoint Aerodynamic Optimization

Published in , 2019

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

Airfoil shape optimization using output-based adapted meshes

Published: June 01, 2017

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

Airfoil shape optimization using output-based adapted meshes

Published: June 01, 2017

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

Airfoil shape optimization using output-based adapted meshes

Published: June 01, 2017

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

Airfoil shape optimization using output-based adapted meshes

Published: June 01, 2017

In this paper, gradient-based aerodynamic shape optimization with output constraints is implemented using adaptive meshes updated via adjoint-based error estimates. All the constraints, including geometry and trim conditions, are handled simultaneously in the optimization. The trim constraints may involve outputs that are not directly targeted for optimization, and hence also not for error estimation and mesh adaptation. However, numerical errors in these outputs often indirectly affect the calculation of the objective. The method adopted in this work takes this effect into account, so that the mesh is adapted to predict both objective outputs and constraint outputs with appropriate accuracy. In other words, the entire optimization problem is targeted for adaptation. Instead of optimizing with a single fixed mesh resolution, the objective function is first evaluated on a relatively coarse mesh, which is subsequently adapted as the shape optimization proceeds. As the shape approaches the optimal design, the mesh becomes finer, in necessary regions, leading to a multi-fidelity optimization process. The multi-fidelity framework saves computational resources by reducing the mesh size at early stages of optimization, when the design is far from optimal and most of the shape changes happen. The optimization at each fidelity terminates once the change of the objective is smaller than the tolerance (estimated error) at the current fidelity mesh, reducing unnecessary effort in optimizing on low-fidelity meshes. The use of output-based error estimates prevents over-optimizing on a coarse mesh, or over-refining on an undesired shape. The proposed framework is demonstrated on several optimization problems, starting with a NACA 0012 airfoil. In each case, the shape is optimized to minimize the drag given a target lift and a minimum airfoil area. The accuracy and efficiency of the proposed method are investigated through comparisons with existing optimization techniques. We expect the framework to be even more important for more complex configurations, dramatic shape changes, or high-accuracy requirements.

Guodong Chen

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