Hyper-reduction over nonlinear manifolds for large nonlinear mechanical systems
Hyper-reduction over nonlinear manifolds for large nonlinear mechanical systems
复制标题
大型非线性机械系统非线性流形的超约简
DOI：
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发表时间：
2017
影响因子：
2
通讯作者：
P. Tiso
中科院分区：
文献类型：
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作者：
Shobhit Jain;P. Tiso
大型非线性有限元离散系统模型降阶的一般趋势是将控制方程伽辽金投影到低维线性子空间上。虽然这减少了系统中未知数的数量，但由于在非线性项的评估中涉及的过高的计算成本，用于获得简化解的计算成本仍然可能很高。然后使用超约化方法对这些非线性项进行快速近似。在有限元背景下，能量守恒采样和加权（ECSW）方法已成为基于Galerkin投影的降阶模型（ROM）超约化的有效工具。模型简化的最新趋势涉及使用非线性流形，这涉及到流形的切空间上的投影。虽然有许多方法来识别这种非线性流形，但在这种ROM中加速计算的超归约技术很少。在这项工作中，我们提出了一个扩展ECSW允许超约简使用非线性映射，同时保持其理想的稳定性和结构保持属性。作为一个概念的证明，建议的超简化技术证明了一个平板和一个现实的机翼结构，其动态已被证明是在一个非线性（二次）流形上发展的模型。采用该方法对机翼结构进行了在线仿真，相对于全系统仿真，在线速度提高了一千倍以上，高于采用ECSW的线性仿真。
Common trends in model reduction of large nonlinear finite element (FE)-discretized systems involve Galerkin projection of the governing equations onto a low-dimensional linear subspace. Though this reduces the number of unknowns in the system, the computational cost for obtaining the reduced solution could still be high due to the prohibitive computational costs involved in the evaluation of nonlinear terms. Hyper-reduction methods are then used for fast approximation of these nonlinear terms. In the finite element context, the energy conserving sampling and weighing (ECSW) method has emerged as an effective tool for hyper-reduction of Galerkin-projection-based reduced-order models (ROMs). More recent trends in model reduction involve the use of nonlinear manifolds, which involves projection onto the tangent space of the manifold. While there are many methods to identify such nonlinear manifolds, hyper-reduction techniques to accelerate computation in such ROMs are rare. In this work, we propose an extension to ECSW to allow for hyper-reduction using nonlinear mappings, while retaining its desirable stability and structure-preserving properties. As a proof of concept, the proposed hyper-reduction technique is demonstrated over models of a flat plate and a realistic wing structure, whose dynamics have been shown to evolve over a nonlinear (quadratic) manifold. An online speed-up of over one thousand times relative to the full system has been obtained for the wing structure using the proposed method, which is higher than its linear counterpart using the ECSW.