In this paper, we extend the structure-preserving interpolatory model reduction framework, originally developed for linear systems, to structured bilinear control systems. Specifically, we give explicit construction formulae for the model reduction bases to satisfy different types of interpolation conditions. First, we establish the analysis for transfer function interpolation for single-input single-output structured bilinear systems. Then, we extend these results to the case of multi-input multi-output structured bilinear systems by matrix interpolation. The effectiveness of our structure-preserving approach is illustrated by means of various numerical examples.
In this paper, we present an interpolation framework for structure-preserving model order reduction of parametric bilinear dynamical systems. We introduce a general setting, covering a broad variety of different structures for parametric bilinear systems, and then provide conditions on projection spaces for the interpolation of structured subsystem transfer functions such that the system structure and parameter dependencies are preserved in the reduced-order model. Two benchmark examples with different parameter dependencies are used to demonstrate the theoretical analysis.
In this study, the standard IEEE 754 2008 and modulo-based floor functions for rounding non-integers have been presented. Their effects on the accuracy of the bilinear interpolation algorithm have been demonstrated. The improved floor uses the modulo operator in an effort to make each non-integer addend an integer when the remainder between the multiplicative inverse of the fractional factor and the numerator is greater than zero. The experiments demonstrated relatively positive effects with the improved floor function and the alternativeness to the standard round function.
In this paper we suggest a moment matching method for quadratic-bilinear dynamical systems. Most system-theoretic reduction methods for nonlinear systems rely on multivariate frequency representations. Our approach instead uses univariate frequency representations tailored towards user-pre-defined families of inputs. Then moment matching corresponds to a one-dimensional interpolation problem, not to multi-dimensional interpolation as for the multivariate approaches, i.e., it also involves fewer interpolation frequencies to be chosen. Comparing to former contributions towards nonlinear model re…
A number of key scientific computing applications that are based upon tensor-product grid constructions, such as numerical weather prediction (NWP) and combustion simulations, require property-preserving interpolation. Essentially non-oscillatory (ENO) interpolation is a classic example of such interpolation schemes. In the aforementioned application areas, property preservation often manifests itself as a requirement for either data boundedness or positivity preservation. For example, in NWP, one may have to interpolate between the grid on which the dynamics is calculated to a grid on which t…
It is well-known that the univariate Multiquadric quasi-interpolation operator is constructed based on the piecewise linear interpolation by |x|. In this paper, we first introduce a new transcendental RBF based on the hyperbolic tangent function as a smooth approximant to f(r)=r with higher accuracy and better convergence properties than the multiquadric. Then Wu-Schaback's quasi-interpolation formula is rewritten using the proposed RBF. It preserves convexity and monotonicity. We prove that the proposed scheme converges with a rate of O(h^2). So it has a higher degree of smoothness. Some nume…
Solution interpolation between deforming meshes is an important component for several applications in scientific computing, including indirect arbitrary-Lagrangian-Eulerian and rezoning moving mesh methods in numerical solution of partial differential equations. In this paper, a high-order, conservative, and positivity-preserving interpolation scheme is developed based on the discontinuous Galerkin solution of a linear time-dependent equation on deforming meshes. The scheme works for bounded but otherwise arbitrary mesh deformation from the old mesh to the new one. The cost and positivity pres…
We examine interpolatory model reduction methods that are well-suited for treating large scale port-Hamiltonian differential-algebraic systems in a way that is able to preserve and indeed, take advantage of the underlying structural features of the system. We introduce approaches that incorporate regularization together with prudent selection of interpolation data. We focus on linear time-invariant systems and present a systematic treatment of a variety of model classes that include combinations of index-$1$ and index-$2$ systems, describing in particular how constraints may be represented in …
An important component of a number of computational modeling algorithms is an interpolation method that preserves the positivity of the function being interpolated. This report describes the numerical testing of a new positivity-preserving algorithm that is designed to be used when interpolating from a solution defined on one grid to different spatial grid. The motivating application for this work was a numerical weather prediction (NWP) code that uses a spectral element mesh discretization for its dynamics core and a cartesian tensor product mesh for the evaluation of its physics routines. Th…
Preserving the positivity of the deformation gradient determinant in intergrid interpolation by combining RBFs and SVD: application to cardiac electromechanics
Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving fr…
We present a new structure-preserving model order reduction (MOR) framework for large-scale port-Hamiltonian descriptor systems (pH-DAEs). Our method exploits the structural properties of the Rosenbrock system matrix for this system class and utilizes condensed forms which often arise in applications and reveal the solution behaviour of a system. Provided that the original system has such a form, our method produces reduced-order models (ROMs) of minimal dimension, which tangentially interpolate the original model's transfer function and are guaranteed to be again in pH-DAE form. This allows t…
In this paper, we consider model order reduction for bilinear systems with non-zero initial conditions. We discuss choices of Gramians for both the homogeneous and the inhomogeneous parts of the system individually and prove how these Gramians characterize the respective dominant subspaces of each of the two subsystems. Proposing different, not necessarily structure preserving, reduced order methods for each subsystem, we establish several strategies to reduce the dimension of the full system. For all these approaches, error bounds are shown depending on the truncated Hankel singular values of…
A structure-preserving discretisation of SO(3)-rotation fields for finite Cosserat micropolar elasticity
University of Luxembourg（卢森堡大学）；University of Exeter（埃克塞特大学）；Technische Universität Berlin（柏林工业大学）；Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；Technische Universität Dortmund（多特蒙德工业大学）
In this paper interpolation of two planar points, corresponding tangent directions and curvatures with Pythagorean-hodograph (PH) curves of degree seven preserving an arc length is considered. A general approach using complex representation of PH curves is presented and a detailed analysis of the problem for data arising from a circular arc is provided. In the case of several solutions some criteria for the selection of the most appropriate one are described and an asymptotic analysis is given. Several numerical examples are included which confirm theoretical results.
