Adaptive numerical modeling using the hierarchical Fup basis functions and control volume isogeometric analysis

Abstract

Summary 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.

Numerical Algorithms for Estimating Probability Density Function Based on the Maximum Entropy Principle and Fup Basis Functions

“…Therefore, the MaxEnt optimization procedure for estimating the probability density function starts from the sixth moment (with the zeroth moment engaged, i.e., ). For numerical operations in the domain to be performed efficiently, it is necessary to modify basis functions that have non-zero values within the domain and the vertices of which are outside the domain by expressing them in the form of a linear combination of the original basis functions [ 50 ]. Figure 3 shows the basis functions in the observed domain for the estimation with six moments ( ).…”

“…The first monographs on the results of research were published in [ 36 , 37 ], while in [ 38 , 39 ] a detailed analysis of the current publications on ABF is provided, from the first publications until now. Furthermore, based on ABF theory, many different classes of weight functions have arisen, which are often used, especially in digital signal processing [ 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 ] and groundwater flow modeling [ 48 , 49 , 50 , 51 ]. A very interesting theory of atomic solitons is based on ABF theory that is used in new areas, like matter quantization, quantum gravity, Higgs fields, unified theory of nature, as well as in non-traditional areas, like medicine and life sciences, geology and financial markets [ 52 , 53 , 54 , 55 ].…”

Atomic Spacetime Model Based on Atomic AString Functions

“…The motivation for this research is to demonstrate how two theories-General Relativity (GR) [13] [14] [15] [16] and Atomic AString Functions [2]- [12]-can be combined as well as to offer a novel mathematical interpretation of spacetime field as a superposition of flexible 'solitonic atoms' (Atomic Solitons). The combined Atomic Spacetime theory is based on formulated Atomization Theorems ( §5) allowing representation of polynomials, analytic functions, and solutions of differential equations of mathematical physics including GR [13]- [33] via superposition of finite Atomic AString Functions resembling flexible quanta ( §3, 7). It leads to novel Spacetime Atomization models and atomic metriants ( §3, 7, 8) as well as offers some variants of unified field theory based on Atomic Solitons ( §8, 9).…”

Full space‐time adaptive method based on collocation strategy and implicit multirate time stepping

“…More about of Fup functions and their application for numerical modeling can be found in References 37‐41. Moreover, our recent publication 42 presents novel approach for spline‐based hp‐adaptivity and contains supplementary materials with detailed mathematical background of the Fup basis functions.…”