scholarly journals Bernstein-Polynomials-Based Highly Accurate Methods for One-Dimensional Interface Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Jiankang Liu ◽  
Zhoushun Zheng ◽  
Qinwu Xu
Keyword(s):  
Bernstein Polynomials ◽  
Jump Condition ◽  
Interface Problems ◽  
Weak Formulation ◽  
Basis Expansion ◽  
One Dimensional ◽  
Lagrange Expansion ◽  
Galerkin Formulation ◽  
Expansion Coefficients ◽  
Lagrange Basis

A new numerical method based on Bernstein polynomials expansion is proposed for solving one-dimensional elliptic interface problems. Both Galerkin formulation and collocation formulation are constructed to determine the expansion coefficients. In Galerkin formulation, the flux jump condition can be imposed by the weak formulation naturally. In collocation formulation, the results obtained by B-polynomials expansion are compared with that obtained by Lagrange basis expansion. Numerical experiments show that B-polynomials expansion is superior to Lagrange expansion in both condition number and accuracy. Both methods can yield high accuracy even with small value ofN.

Finite Element Models of One Dimensional Flows With Node-Dependent Accuracy

2018 ◽  
Author(s):  
Daniele Guarnera ◽  
Enrico Zappino ◽  
Alfonso Pagani ◽  
Erasmo Carrera
Keyword(s):  
Finite Element ◽  
Stokes Problem ◽  
Computational Cost ◽  
Circulatory System ◽  
Element Model ◽  
Laminar Flows ◽  
Sudden Expansion ◽  
Fe Modelling ◽  
Weak Formulation ◽  
One Dimensional

The formulation of simplified models in the description of flow fields can be highly interesting in many complex network such as the circulatory system. This work presents a refined one-dimensional finite element model with node-dependent kinematics applied to incompressible and laminar flows. In the framework of 1D-FE modelling, this methodology is a new development of the Carrera Unified Formulation (CUF), which is largely employed in structural mechanics. According to the CUF, the weak formulation of the Stokes problem is expressed in terms of fundamental nuclei and, in this novel implementation, the kinematics can be defined node by node, realizing different levels of refinements within the main direction of the pipe. Such feature allows to increase the accuracy of the model only in the areas of the domain where it is required, i.e. particular boundary condition, barriers or sudden expansion. Some typical CFD examples are proposed to validate this novel technique, including Stokes flows in uniform and non-uniform domains. For each numerical example, different combinations of 1D models have been considered to account for different kinematic approximations of flows, and in particular, models based on Taylor and Lagrange expansion have been used. The results, compared with ones obtained from uniform kinematics 1D models and with those come from available tools, highlight the capability of the proposed model in handling non-conventional boundary conditions with ease and in preserving the computational cost without any accuracy loss.

scholarly journals An efficient numerical method for one-dimensional hyperbolic interface problems

2019 ◽  
Vol 29 ◽  
pp. 01002
Author(s):  
Chartese Jones ◽  
Xu Zhang
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Method Of Lines ◽  
Interface Problems ◽  
Uniform Mesh ◽  
One Dimensional ◽  
Immersed Finite Element ◽  
New Methods ◽  
The Method Of Lines

In this paper, we develop an efficient numerical scheme for solving one-dimensional hyperbolic interface problems. The immersed finite element (IFE) method is used for spatial discretization, which allows the solution mesh to be independent of the interface. Consequently, a fixed uniform mesh can be used throughout the entire simulation. The method of lines is used for temporal discretization. Numerical experiments are provided to show the features of these new methods.

scholarly journals The effects of numerical resolution, heating timescales and background heating on thermal non-equilibrium in coronal loops

2019 ◽  
Vol 625 ◽  
pp. A149 ◽  
Author(s):  
C. D. Johnston ◽  
P. J. Cargill ◽  
P. Antolin ◽  
A. W. Hood ◽  
I. De Moortel ◽  
...  
Keyword(s):  
Approximate Method ◽  
Computational Cost ◽  
Jump Condition ◽  
Coronal Loops ◽  
Hydrodynamic Models ◽  
Plasma Properties ◽  
One Dimensional ◽  
Numerical Resolution ◽  
Good Agreement ◽  
Non Equilibrium

Thermal non-equilibrium (TNE) is believed to be a potentially important process in understanding some properties of the magnetically closed solar corona. Through one-dimensional hydrodynamic models, this paper addresses the importance of the numerical spatial resolution, footpoint heating timescales and background heating on TNE. Inadequate transition region (TR) resolution can lead to significant discrepancies in TNE cycle behaviour, with TNE being suppressed in under-resolved loops. A convergence on the periodicity and plasma properties associated with TNE required spatial resolutions of less than 2 km for a loop of length 180 Mm. These numerical problems can be resolved using an approximate method that models the TR as a discontinuity using a jump condition, as proposed by Johnston et al. (2017a, A&A, 597, A81; 2017b, A&A, 605, A8). The resolution requirements (and so computational cost) are greatly reduced while retaining good agreement with fully resolved results. Using this approximate method we (i) identify different regimes for the response of coronal loops to time-dependent footpoint heating including one where TNE does not arise and (ii) demonstrate that TNE in a loop with footpoint heating is suppressed unless the background heating is sufficiently small. The implications for the generality of TNE are discussed.

scholarly journals A Cartesian Grid Method for Modeling Charge Distribution on Interfaces via Augmented Technique

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Haifeng Ji ◽  
Qian Zhang ◽  
Bin Zhang
Keyword(s):  
Charge Distribution ◽  
Grid Method ◽  
Jump Condition ◽  
Cartesian Grid ◽  
Interface Problems ◽  
Interface Problem ◽  
Jump Conditions ◽  
Cartesian Grid Method ◽  
Electric Quantity ◽  
The Given

In the electrostatic field computations, second-order elliptic interface problems with nonhomogeneous interface jump conditions need to be solved. In realistic applications, often the total electric quantity on the interface is given. However, the charge distribution on the interface corresponding to the nonhomogeneous interface jump condition is unknown. This paper proposes a Cartesian grid method for solving the interface problem with the given total electric quantity on the interface. The proposed method employs both the immersed finite element with the nonhomogeneous interface jump condition and the augmented technique. Numerical experiments are presented to show the accuracy and efficiency of the proposed method.

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