Jeff Borggaard

Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison

This paper puts forth two new closure models for the proper orthogonal decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model. These models, which are considered state-of-the-art in large eddy simulation, together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of 3D turbulent flow past a circular cylinder at Re=1000. Five criteria are used to judge the performance of the proper orthogonal decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the POD coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.

Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition

The proper orthogonal decomposition (POD) is the prevailing method for basis generation in the model reduction of fluids. A serious limitation of this method, however, is that it is empirical. In other words, this basis accurately represents the flow data used to generate it, but may not be accurate when applied ‘off-design’. Thus, the reduced-order model may lose accuracy for flow parameters (e.g. Reynolds number, initial or boundary conditions and forcing parameters) different from those used to generate the POD basis and generally does. This paper investigates the use of sensitivity analysis in the basis selection step to partially address this limitation. We examine two strategies that use the sensitivity of the POD modes with respect to the problem parameters. Numerical experiments performed on the flow past a square cylinder over a range of Reynolds numbers demonstrate the effectiveness of these strategies. The newly derived bases allow for a more accurate representation of the flows when exploring the parameter space. Expanding the POD basis built at one state with its sensitivity leads to low-dimensional dynamical systems having attractors that approximate fairly well the attractor of the full-order Navier–Stokes equations for large parameter changes.

Two-level discretizations of nonlinear closure models for proper orthogonal decomposition

Proper orthogonal decomposition has been successfully used in the reduced-order modeling of complex systems. Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed. Since modern closure models for turbulent flows are generally nonlinear, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This paper proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced-order models. The two-level method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter @n=10^-^3, the two-dimensional flow past a cylinder at Reynolds number Re=200, and the three-dimensional flow past a cylinder at Reynolds number Re=1000.

Boundary element implicit differentiation equations for design sensitivities of axisymmetric structures

Abstract Design sensitivity analysis of axisymmetric elastic media is formulated using boundary elements. The kernels for sensitivity matrices are obtained through implicit differentiation of the corresponding boundary element elasticity kernels. The singular terms are obtained by applying the boundary displacements and tractions, and their respective sensitivities for the rigid body motion mode and the inflation mode. The equations for the recovery of sensitivities of axisymmetric boundary stresses are presented. As a check on accuracy, the approach is applied to a series of examples for which analytical elasticity solutions are available. The predictions for both displacement- and stress-sensitivities are accurate. Additional examples are provided to demonstrate the versatility of the present approach.

A SENSITIVITY EQUATION APPROACH TO SHAPE OPTIMIZATION IN FLUID FLOWS

In this paper we apply a sensitivity equation method to shape optimization problems. An algorithm is developed and tested on a problem of designing optimal forebody simulators for a 2D, inviscid supersonic flow. The algorithm uses a BFGS/Trust Region optimization scheme with sensitivities computed by numerically approximating the linear partial differential equations that determine the flow sensitivities. Numerical examples are presented to illustrate the method.

A CONTINUOUS SENSITIVITY EQUATION APPROACH TO OPTIMAL DESIGN IN MIXED CONVECTION

In this paper, we consider the design of a mixed convection system to achieve optimum heat transfer properties. Our design methodology consists of using an adaptive finite element method to obtain approximations for both flow and sensitivity variables. These quantities allow us to calculate values of a design objective function and its gradient. A BFGS/trust-region optimization algorithm uses this information to find optimum parameter values. The adaptive remeshing strategy is constructed to ensure accurate resolution of both flow and sensitivity variables throughout the design iterations. We present this methodology along with numerical results that include a validation of the effectiveness of our remeshing strategy along with the solution to an optimal design problem.In this paper, we consider the design of a mixed convection system to achieve optimum heat transfer properties. Our design methodology consists of using an adaptive finite element method to obtain approximations for both flow and sensitivity variables. These quantities allow us to calculate values of a design objective function and its gradient. A BFGS/trust-region optimization algorithm uses this information to find optimum parameter values. The adaptive remeshing strategy is constructed to ensure accurate resolution of both flow and sensitivity variables throughout the design iterations. We present this methodology along with numerical results that include a validation of the effectiveness of our remeshing strategy along with the solution to an optimal design problem.

A GENERAL CONTINUOUS SENSITIVITY EQUATION FORMULATION FOR COMPLEX FLOWS

In this article, we develop a general formulation of the continuous sensitivity equation method that accounts for a complex parameter dependence in both flow variables and physical fluid properties (such as viscosity, thermal conductivity, etc.). This formulation unifies the treatment of shape sensitivities and value sensitivities. The result leads to the development of software that is suitable for a wide range of problems. In addition to details of an implementation within an existing adaptive finite-element program, we perform a careful verification study as well as demonstrate the flexibility of the software by computing cost function gradients for an optimization algorithm.

On Efficient Solutions to the Continuous Sensitivity Equation Using Automatic Differentiation

Shape sensitivity analysis is a tool that provides quantitative information about the influence of shape parameter changes on the solution of a partial differential equation (PDE). These shape sensitivities are described by a continuous sensitivity equation (CSE). Automatic differentiation (AD) can be used to perform this sensitivity analysis without writing any additional code to solve the sensitivity equation. The approximate solution of the PDE uses a spatial discretization (mesh) that often depends on the shape parameters. Therefore, the straightforward application of AD introduces derivatives of the mesh. There are two drawbacks to this approach. First, extra computational effort (especially memory) is used in these calculations due to mesh sensitivities. Second, this mesh sensitivity information needs to be computed in order to obtain accurate results. In this work, we provide a methodology that avoids mesh sensitivities (and their drawbacks) by defining a modified PDE on a fixed domain (i.e., independent of the shape parameter) such that AD provides the desired approximation of the CSE. Using two examples, we demonstrate significant improvement in the computational effort, both in terms of floating point operations and memory requirements. We explain how these code modifications can be applied to a wide variety of practical problems with minimal changes to the original code. These changes are negligible when compared to the complexity of writing a separate solver for the sensitivity equation.

Control, estimation and optimization of energy efficient buildings

Commercial buildings are responsible for a significant fraction of the energy consumption and greenhouse gas emissions in the U.S. and worldwide. Consequently, the design, optimization and control of energy efficient buildings can have a tremendous impact on energy cost and greenhouse gas emission. Buildings are complex, multi-scale in time and space, multi-physics and highly uncertain dynamic systems with wide varieties of disturbances. Recent results have shown that by considering the whole building as an integrated system and applying modern estimation and control techniques to this system, one can achieve greater efficiencies than obtained by optimizing individual building components such as lighting and HVAC. We consider estimation and control for a distributed parameter model of a multi-room building. In particular, we show that distributed parameter control theory, coupled with high performance computing, can provide insight and computational algorithms for the optimal placement of sensors and actuators to maximize observability and controllability. Numerical examples are provided to illustrate the approach. We also discuss the problems of design and optimization (for energy and CO2 reduction) and control (both local and supervisory) of whole buildings and demonstrate how sensitivities can be used to address these problems.

A General Continuous Sensitivity Equation Formulation for the k-ε Model of Turbulence

In this paper, we develop a general formulation of the continuous sensitivity equations (CSEs) for the standard model of turbulence with wall functions. The development is performed for value parameters that do not affect the geometry of the computational domain. The formulation accounts for complex parameter dependencies and results in the development of software that is suitable for a wide range of problems. In addition to details of an implementation within an existing adaptive finite element program, we perform a careful verification study and present an application of sensitivity analysis to turbulent flow over a flat plate.