Application of the Solution Structure Method in Numerically Solving Poisson’s Equation on the Basis of Atomic Functions

Kozulić, Vedrana; Gotovac, Blaž

doi:10.1142/s0219876218500330

Cited by 3 publications

(3 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The presented method in this work is not classical IGA, which mostly uses B-spline or NURBS basis function in conjunction with the Galerkin or collocation formulation. Rather, here, we use Fup basis functions [44][45][46] and the control-volume formulation and refer to the method as control-volume isogeometric analysis (CV-IGA). Fup basis functions belong to the class of atomic basis functions and can be regarded as infinitely differentiable B-splines.…”

Section: The Numerical Modelmentioning

confidence: 99%

Groundwater Flow Modeling in Karst Aquifers: Coupling 3D Matrix and 1D Conduit Flow via Control Volume Isogeometric Analysis—Experimental Verification with a 3D Physical Model

Malenica

Gotovac

Kamber

et al. 2018

Water

View full text Add to dashboard Cite

A novel numerical model for groundwater flow in karst aquifers is presented. A discrete-continuum (hybrid) approach, in which a three-dimensional matrix flow is coupled with a one-dimensional conduit flow, was used. The laminar flow in the karst matrix is described by a variably saturated flow equation to account for important hydrodynamic effects in both the saturated and unsaturated zones. Turbulent conduit flow for both free surface and pressurized flow conditions was captured via the noninertia wave equation, whereas the coupling of two flow domains was established through an exchange term proportional to head differences. The novel numerical approach based on Fup basis functions and control-volume formulation enabled us to obtain smooth and locally conservative numerical solutions. Due to its similarity to the isogeometric analysis concept (IGA), we labeled it as control-volume isogeometric analysis (CV-IGA). Since realistic verification of the karst flow models is an extremely difficult task, the particular contribution of this work is the construction of a specially designed 3D physical model ( dimensions: 5.66 × 2.95 × 2.00 m) in order to verify the developed numerical model under controlled laboratory conditions. Heterogeneous porous material was used to simulate the karst matrix, and perforated pipes were used as karst conduits. The model was able to capture many flow characteristics, such as the interaction between the matrix and conduit, rainfall infiltration through the unsaturated zone, direct recharge through sinkholes, and both free surface and pressurized flow in conduits. Two different flow experiments are presented, and comparison with numerical results confirmed the validity of the developed karst flow model under complex laboratory conditions.

show abstract

Section: The Numerical Modelmentioning

confidence: 99%

Groundwater Flow Modeling in Karst Aquifers: Coupling 3D Matrix and 1D Conduit Flow via Control Volume Isogeometric Analysis—Experimental Verification with a 3D Physical Model

Malenica

Gotovac

Kamber

et al. 2018

Water

View full text Add to dashboard Cite

show abstract

“…So Fup basis functions are only one possible candidate for such methodology and other choices such as B‐splines, wavelets, or radial basis functions can be used among others. 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.…”

Section: Introductionmentioning

confidence: 99%

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

Malenica

Gotovac

2021

Numerical Methods in Fluids

View full text Add to dashboard Cite

In this paper, we present a full space‐time Adaptive Fup Collocation Method (AFCM) for solving initial‐boundary value problems with particular application on advection‐dominated advection‐diffusion problems. The spatial adaptive strategy dynamically changes the computational grid at each global time step, while the adaptive time‐marching algorithm uses different local time steps for different collocation points based on temporal accuracy criteria. Contrary to the existing space‐time adaptive methods that are based on explicit temporal discretization and scale‐dependent local time stepping, the presented method is implicit and resolves all spatial and temporal scales independently of each other. This means that smaller local time steps are used only for spatial zones where temporal solution changes are intensive, which are not necessarily the zones with finest spatial discretization. In this manner, the computationally efficient full space‐time adaptive strategy accurately resolves small‐scale features and controls numerical error and spurious oscillations. The efficiency and accuracy of AFCM are verified with some classical one‐ and two‐dimensional benchmark test cases.

show abstract

“…Efficient numerical modeling using spline functions does not always have to be associated exclusively with IGA involving geometry transformations, because everything can only be performed in the physical domain. Furthermore, the geometry constraints and boundary conditions can be satisfied exactly using the Rvachev solution structure method (see Rvachev et al [18], Höllig et al [19], and Kozulić and Gotovac [20]).…”

Section: Cases Of Modified Boundary Fupmentioning

confidence: 99%

Adaptive Numerical Modeling of Engineering Problems Using Hierarchical Fup Basis Functions and Control Volume IsoGeometric Analysis

Kamber¹

View full text Add to dashboard Cite

The main objective of this thesis is to utilize the powerful approximation properties of Fup basis functions for numerical solutions of engineering problems with highly localized steep gradients while controlling spurious numerical oscillations and describing different spatial scales. The concept of isogeometric analysis (IGA) is presented as a unified framework for multiscale representation of the geometry and solution. This fundamentally high-order approach enables the description of all fields as continuous and smooth functions by using a linear combination of spline basis functions. Classical IGA usually employs Galerkin or collocation approach using B-splines or NURBS as basis functions. However, in this thesis, a third concept in the form of control volume isogeometric analysis (CV-IGA) is used with Fup basis functions which represent infinitely smooth splines. Novel hierarchical Fup (HF) basis functions is constructed, enabling a local hp-refinement such that they can replace certain basis functions at one resolution level with new basis functions at the next resolution level that have a smaller length of the compact support (h-refinement), but also higher order (p-refinement). This hp-refinement property enables spectral convergence which is significant improvement in comparison to the hierarchical truncated B-splines which enable h-refinement and polynomial convergence. Thus, in domain zones with larger gradients, the algorithm uses smaller local spatial scales, while in other region, larger spatial scales are used, controlling the numerical error by the prescribed accuracy. The efficiency and accuracy of the adaptive algorithm is verified with some classic 1D and 2D benchmark test cases with application to the engineering problems with highly localized steep gradients and advection-dominated problems.

show abstract

Application of the Solution Structure Method in Numerically Solving Poisson’s Equation on the Basis of Atomic Functions

Cited by 3 publications

References 17 publications

Groundwater Flow Modeling in Karst Aquifers: Coupling 3D Matrix and 1D Conduit Flow via Control Volume Isogeometric Analysis—Experimental Verification with a 3D Physical Model

Groundwater Flow Modeling in Karst Aquifers: Coupling 3D Matrix and 1D Conduit Flow via Control Volume Isogeometric Analysis—Experimental Verification with a 3D Physical Model

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

Adaptive Numerical Modeling of Engineering Problems Using Hierarchical Fup Basis Functions and Control Volume IsoGeometric Analysis

Contact Info

Product

Resources

About