1,307 research outputs found

    Passivity-preserving parameterized model order reduction using singular values and matrix interpolation

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    We present a parameterized model order reduction method based on singular values and matrix interpolation. First, a fast technique using grammians is utilized to estimate the reduced order, and then common projection matrices are used to build parameterized reduced order models (ROMs). The design space is divided into cells, and a Krylov subspace is computed for each cell vertex model. The truncation of the singular values of the merged Krylov subspaces from the models located at the vertices of each cell yields a common projection matrix per design space cell. Finally, the reduced system matrices are interpolated using positive interpolation schemes to obtain a guaranteed passive parameterized ROM. Pertinent numerical results validate the proposed technique

    A semi-Lagrangian Vlasov solver in tensor train format

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    In this article, we derive a semi-Lagrangian scheme for the solution of the Vlasov equation represented as a low-parametric tensor. Grid-based methods for the Vlasov equation have been shown to give accurate results but their use has mostly been limited to simulations in two dimensional phase space due to extensive memory requirements in higher dimensions. Compression of the solution via high-order singular value decomposition can help in reducing the storage requirements and the tensor train (TT) format provides efficient basic linear algebra routines for low-rank representations of tensors. In this paper, we develop interpolation formulas for a semi-Lagrangian solver in TT format. In order to efficiently implement the method, we propose a compression of the matrix representing the interpolation step and an efficient implementation of the Hadamard product. We show numerical simulations for standard test cases in two, four and six dimensional phase space. Depending on the test case, the memory requirements reduce by a factor 10210310^2-10^3 in four and a factor 10510610^5-10^6 in six dimensions compared to the full-grid method

    Dimensional hyper-reduction of nonlinear finite element models via empirical cubature

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    We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accuracy.Peer ReviewedPostprint (published version

    Physics-based passivity-preserving parameterized model order reduction for PEEC circuit analysis

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    The decrease of integrated circuit feature size and the increase of operating frequencies require 3-D electromagnetic methods, such as the partial element equivalent circuit (PEEC) method, for the analysis and design of high-speed circuits. Very large systems of equations are often produced by 3-D electromagnetic methods, and model order reduction (MOR) methods have proven to be very effective in combating such high complexity. During the circuit synthesis of large-scale digital or analog applications, it is important to predict the response of the circuit under study as a function of design parameters such as geometrical and substrate features. Traditional MOR techniques perform order reduction only with respect to frequency, and therefore the computation of a new electromagnetic model and the corresponding reduced model are needed each time a design parameter is modified, reducing the CPU efficiency. Parameterized model order reduction (PMOR) methods become necessary to reduce large systems of equations with respect to frequency and other design parameters of the circuit, such as geometrical layout or substrate characteristics. We propose a novel PMOR technique applicable to PEEC analysis which is based on a parameterization process of matrices generated by the PEEC method and the projection subspace generated by a passivity-preserving MOR method. The proposed PMOR technique guarantees overall stability and passivity of parameterized reduced order models over a user-defined range of design parameter values. Pertinent numerical examples validate the proposed PMOR approach
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