Model reduction of parabolic PDEs using multivariate splines

Abstract

A new methodology is presented for model reduction of linear parabolic partial differential equations (PDEs) on general geometries using multivariate splines on triangulations. State-space descriptions are derived that can be used for control design. This method uses Galerkin projection with B-splines to derive a finite set of ordinary differential equations (ODEs). Any desired smoothness conditions between elements as well as the boundary conditions are flexibly imposed as a system of side constraints on the set of ODEs. Projection of the set of ODEs on the null space of the system of side constraints naturally produces a reduced-order model that satisfies these constraints. This method can be applied for both in-domain control and boundary control of parabolic PDEs with spatially varying coefficients on general geometries. The reduction method is applied to design and implement feedback controllers for stabilisation of a 1-D unstable heat equation and a more challenging 2-D reaction-convection-diffusion equation on an irregular domain. It is shown that effective feedback stabilisation can be achieved using low-order control models.

Localised estimation and control of linear instabilities in two-dimensional wall-bounded shear flows

“…For simulation and control design a finite dimensional representation of (2.22) is required. In Tol et al (2016) a framework is presented for deriving state-space descriptions for a general class of linear parabolic PDEs to which standard control theoretic tools can be applied. This method is also used in this work and uses multivariate B-splines of arbitrary degree and smoothness defined on triangulations (Farin 1986;de Boor 1987;Lai & Schumaker 2007) to find matrix representations of all operators in (2.21).…”

“…This basis is used to spatially discretise the system. The resulting discrete system is transformed to statespace format using a null space projection method (Tol et al 2016). This projection employs a similar state transformation as in (2.17), but in a discrete setting, and results in a reduced number of states that have a minimal non-zero support for the smooth divergence free spline space S ⊂ X .…”

“…In appendix A the numerical scheme to derive the finite dimensional state-space system of the flow is described. This numerical scheme is based on multivariate B-splines defined on triangulations (Farin 1986;de Boor 1987;Lai & Schumaker 2007) and is an extension of the model reduction scheme for parabolic PDEs presented in Tol et al (2016) to fluid flows. In Appendix B the state-space formulas for the controller that solves the H 2 optimal control problem are given.…”

“…In Awanou & Lai (2004) a numerical scheme is presented for approximating steady Navier-Stokes equations in velocity pressure formulation using multivariate splines. This numerical scheme is combined with the framework from Tol et al (2016) to derive state descriptions for the linearised Navier-Stokes equations and is presented A.1. The state-space system for the case of the non-periodic channel flow was validated in section 3 by comparing the spatial stability with the predictions from LST.…”

“…This method uses multivariate splines defined on triangulation's (Farin 1986;de Boor 1987;Lai & Schumaker 2007) to find matrix representations of all operators in (2.21) and applies to general geometries and general control configurations. This method is an extension of the model reduction scheme for parabolic PDEs presented by Tol et al (2016) to fluid flows. In Awanou & Lai (2004) a numerical scheme is presented for approximating steady Navier-Stokes equations in velocity pressure formulation using multivariate splines.…”