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A Krylov enhanced proper orthogonal decomposition method for efficient nonlinear model reduction

机译:有效压缩非线性模型的Krylov增强固有正交分解方法

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摘要

A model order reduction method is proposed for approximating nonlinear partial differential equations (PDEs). The method attempts to combine the desirable attributes of Krylov reduction and proper orthogonal decomposition (POD) and is entitled Krylov enhanced POD (KPOD). The method approximates nonlinear input/output (I/O) behavior using a sequence of state dependent Krylov subspaces which are obtained via simulation of a nonlinear finite element discretized system. POD is then implemented to extract, from the sequence of subspaces, a projection basis that describes the state dependent variance of the Krylov subspaces. Due to Krylov's moment matching property, the variance describes how the I/O of a sequence of Krylov reduced models would vary as the original model's state varies. Galerkin projection is performed using the extracted basis to yield a low dimensional I/O approximation of the original nonlinear model. Reduced order models of an electro-thermal switch generated using the conventional POD and KPOD are presented and used to compare the I/O approximation capabilities of each method. Our results indicate that KPOD reduced models are capable of accurately modeling I/O using a sequence of Krylov subspaces generated from a single nonlinear discretized model simulation while POD can require multiple nonlinear discretized model simulations to generate a reduced model with similar I/O accuracy. The proposed method can be a potential asset for practical applications in the area of I/O behavior prediction and design optimization of nonlinear systems.
机译:提出了一种模型阶数约简方法来逼近非线性偏微分方程(PDE)。该方法尝试将Krylov约简和适当的正交分解(POD)的理想属性结合起来,称为Krylov增强POD(KPOD)。该方法使用一系列依赖状态的Krylov子空间来近似非线性输入/输出(I / O)行为,该子空间是通过非线性有限元离散系统的仿真获得的。然后实现POD,以从子空间序列中提取描述Krylov子空间的状态相关方差的投影基础。由于Krylov的矩匹配特性,方差描述了一系列Krylov简化模型的I / O将如何随着原始模型状态的变化而变化。使用提取的基础执行Galerkin投影,以产生原始非线性模型的低维I / O近似值。介绍了使用常规POD和KPOD生成的电热开关的降阶模型,并将其用于比较每种方法的I / O逼近能力。我们的结果表明,KPOD精简模型能够使用从单个非线性离散化模型仿真生成的一系列Krylov子空间对I / O进行精确建模,而POD可能需要多个非线性离散化模型仿真以生成具有相似I / O精度的精简模型。所提出的方法可能是I / O行为预测和非线性系统设计优化领域中实际应用的潜在资产。

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