Adaptive numerical modeling using the hierarchical Fup basis functions and control volume isogeometric analysis,International Journal for Numerical Methods in Fluids

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Adaptive numerical modeling using the hierarchical Fup basis functions and control volume isogeometric analysis
International Journal for Numerical Methods in Fluids ( IF 1.7 ) Pub Date : 2020-04-07 , DOI: 10.1002/fld.4830
Grgo Kamber ₁ , Hrvoje Gotovac ₁ , Vedrana Kozulić ₁ , Luka Malenica ₁ , Blaž Gotovac ₁

Affiliation

A novel adaptive algorithm that is based on new hierarchical Fup (HF) basis functions and a control volume formulation is presented. Because of its similarity to the concept of isogeometric analysis (IGA), we refer to it as control volume isogeometric analysis (CV‐IGA). Among other interesting properties, the IGA introduced k‐refinement as advanced version of hp‐refinement, where every basis function of the nth order from one resolution level are replaced by a linear combination of more basis functions of the n+1th order at the next resolution level. However, k‐refinement can be performed only on whole domain, while local adaptive k‐refinement is not possible with classical B‐spline basis functions. HF basis functions (infinitely differentiable splines) satisfy partition of unity, and they are linearly independent and locally refinable. Their main feature is execution of the adaptive local hp‐refinement because any basis function of the nth order from one resolution level can be replaced by a linear combination of more basis functions of the n+1th order at the next resolution level providing spectral convergence order. The comparison between uniform vs hierarchical adaptive solutions is demonstrated, and it is shown that our adaptive algorithm returns the desired accuracy while strongly improving the efficiency and controlling the numerical error. In addition to the adaptive methodology, a stabilization procedure is applied for advection‐dominated problems whose numerical solutions “suffer” from spurious oscillations. Stabilization is added only on lower resolution levels, while higher resolution levels ensure an accurate solution and produce a higher convergence order. Since the focus of this article is on developing HF basis functions and adaptive CV‐IGA, verification is performed on the stationary one‐dimensional boundary value problems.

中文翻译：

使用分层Fup基函数和控制体积等几何分析的自适应数值建模

提出了一种新的自适应算法，该算法基于新的分层Fup（HF）基函数和控制量公式。由于它与等几何分析（IGA）的概念相似，我们将其称为控制体积等几何分析（CV-IGA）。在其他有趣的性质，在IGA引入ķ -refinement先进版本的马力-refinement，其中的每一个基函数ñ从一个分辨率级别阶被的的更多的基函数的线性组合代替Ñ在+ 1顺序下一个分辨率级别。但是，k细化只能在整个域上执行，而局部自适应k经典B样条基函数无法进行优化。HF基函数（可无限微分的样条曲线）满足单位的划分，并且它们是线性独立且可局部优化的。其主要特征是自适应局部的执行马力-refinement因为任何基函数ñ阶从一个分辨率级别可通过的的多个基函数的线性组合来代替Ñ下一分辨率级别的+1阶提供频谱收敛阶。证明了统一和分层自适应解决方案之间的比较，结果表明我们的自适应算法在提高效率和控制数值误差的同时，返回了所需的精度。除自适应方法外，稳定程序适用于对流占主导地位的问题，其数值解因虚假振荡而受苦。仅在较低的分辨率级别上添加了稳定功能，而较高的分辨率级别则确保了准确的解决方案并产生了较高的收敛阶数。由于本文的重点是开发HF基函数和自适应CV-IGA，因此对平稳的一维边界值问题进行了验证。

更新日期：2020-04-07

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