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Multiresolution based adaptive schemes for second order hyperbolic PDEs in elastodynamic problems

机译：弹性力学问题中基于多分辨率的二阶双曲型PDE的自适应方案

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摘要

An enhanced interpolation wavelet-based adaptive-grid scheme is implemented for simulating high gradient smooth solutions (as well as, discontinuous ones) in elastodynamic problems in domains with irregular boundary shapes. In the method, spatially adaptive smoothing is used to improve interpolation property of the solution in high gradient zones. In hyperbolic systems, in fact, there are no certain inherent regularities; hence, the erroneous adapted grid may be achieved because of small spurious oscillations in the solution domain. These oscillations, mainly formed in the vicinity of high gradient and discontinuity zones, make the adaptation procedure strongly unstable. To cover this drawback, enhanced smoothing splines are used to denoise directly non-physical oscillations in the irregular grid points, a kind of ill-posed problem. Controllable smoothing is achieved using non-uniform weight coefficients. As the smoothing splines are a kind of the Thikhonov regularization method, they work stably in irregular grid points. Regarding the Thikhonov regularization method, L-curve scheme could be used to investigate trade-off between accuracy and smoothness of the solutions. This relationship, in fact, could not be reliably captured by common computational methods. The proposed method, in general, is easy and conceptually straightforward; as all calculations are carried out in the physical domain. This concept is verified using a variety of 2D numerical examples.
机译：实现了一种基于增强插值小波的自适应网格方案，用于模拟边界形状不规则的区域中弹性动力学问题中的高梯度平滑解（以及不连续解）。在该方法中，使用空间自适应平滑来改善高梯度区域中解决方案的插值特性。实际上，在双曲系统中，没有某些固有规律。因此，由于解域中的小寄生振荡，可能会导致错误的自适应网格。这些振荡主要在高梯度和不连续区附近形成，使适应过程变得非常不稳定。为了弥补这个缺点，增强的平滑样条用于直接消除不规则网格点中的非物理振动，这是一种不适定的问题。使用不均匀的权重系数可以实现可控的平滑。由于平滑样条线是Thikhonov正则化方法的一种，因此它们可以在不规则网格点上稳定运行。关于Thikhonov正则化方法，可以使用L曲线方案来研究解决方案的精度和平滑度之间的权衡。实际上，这种关系不能通过常用的计算方法可靠地捕获。总体而言，所提出的方法是简单且概念上简单的。因为所有计算都是在物理域中进行的。使用各种2D数值示例验证了此概念。

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