Irina Kalashnikova

Stable Galerkin reduced order models for linearized compressible flow

The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transformation leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using a piecewise-smooth finite element basis. Stability of the reduced order model using this approach is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation.

Galerkin reduced order models for compressible flow with structural interaction

The Galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations. The numerical stability of Galerkin models is shown to depend on the choice of inner product for the projection. For the linearized Euler equations, a symmetry transform leads to a stable formulation for the inner product. Boundary conditions for compressible flow that preserve stability of the reduced order model are constructed. Coupling with a linearized structural dynamics model is made possible through the solid wall boundary condition. Preservation of stability for the discrete implementation of the Galerkin projection is made possible using piecewise-smooth finite element bases. Stability of the coupled fluid/structure system is examined for the case of uniform flow past a thin plate. Stability of the reduced order model for the fluid is demonstrated on several model problems, where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation.

Stable and Efficient Galerkin Reduced Order Models for Non-Linear Fluid Flow.

An efficient model reduction technique for non-linear compressible flow equations is proposed. The approach is based on the continuous Galerkin projection approach, in which the continuous governing equations are projected onto the reduced basis modes in a continuous inner product. It is an extension of the provablystable model reduction methodology developed previously 1‐3, 13 for the linearized compressible flow equations to the non-linear counterparts of these equations. Attention is focussed on two challenges that arise in developing reduced order models (ROMs) for the full Navier-Stokes equations: stability and efficiency. The former challenge is addressed through the introduction of a transformation into the so-called “entropy variables”. It is shown that performing the Galerkin projection step of the model reduction procedure in these variables leads to a ROM that obeys a priori the second law of thermodynamics, or Clausius-Duhem inequality. In this way, the ROM preserves an essential stability property of the governing equations, that of non-decreasing entropy in the solution. Although the discussion assumes that the reduced basis is constructed via the proper orthogonal decomposition (POD), the entropy stability guarantee holds for any choice of reduced basis, not only the POD basis. The challenge of ensuring that the model reduction technique is efficient in the presence of non-linearities is addressed using the “best points” interpolation method (BPIM) of Peraire, Nguyen et al. 16, 17 To help gauge the viability of the proposed model reduction, some preliminary numerical studies are performed on two non-linear scalar conservation laws whose solutions possess inherently non-linear features, such as shocks and rarefactions: the Burgers equation and the Buckley-Leverett equation.

Preconditioner and Convergence Study for the Quantum Computer Aided Design (QCAD) Nonlinear Poisson Problem Posed on the Ottawa Flat 270 Design Geometry

A numerical study aimed to evaluate different preconditioners within the Trilinos Ifpack and ML packages for the Quantum Computer Aided Design (QCAD) non-linear Poisson problem implemented within the Albany code base and posed on the Ottawa Flat 270 design geometry is performed. This study led to some new development of Albany that allows the user to select an ML preconditioner with Zoltan repartitioning based on nodal coordinates, which is summarized. Convergence of the numerical solutions computed within the QCAD computational suite with successive mesh refinement is examined in two metrics, the mean value of the solution (an L{sup 1} norm) and the field integral of the solution (L{sup 2} norm).