Presentation on theme: "Scope Introduction Local radial basis function collocation method (LRBFCM) The adaptation algorithm The refinement criteria The refinement algorithm Numerical."— Presentation transcript:
Scope Introduction Local radial basis function collocation method (LRBFCM) The adaptation algorithm The refinement criteria The refinement algorithm Numerical examples –2D diffusion equation –Convection dominated trans port in circular velocity field Conclusions and further work
Overview of our publications - RBFCM I.Kovačević, A.Poredoš, B.Šarler, Numerical heat Transfer B (2003) (Stefan problem) B.Šarler, J.Perko, D.Gobin, B.Goyeau, H.Power, Int.J.Numer.Methods Heat & Fluid Flow (2004) (Porous media flow – Darcy model) B.Šarler, Engineering Analysis with Boundary Elements (2004) (Porous media flow – Darcy-Brinkman model) B. Šarler, Computer Modeling in Engineering and Sciences (2005) (Navier Stokes) Problems due to ill-conditioning of the collocation matrix for more than approx nodes. Introduction
Overview of our recent publications - LRBFCM B.Šarler, R.Vertnik, Computers & Mathematics with Applications (2006) (Diffusion) R.Vertnik, B.Šarler, Int.J.Numer.Methods Heat & Fluid Flow (2006) (Convection-diffusion & phase change) R.Vertnik, M.Založnik, B.Šarler, Eng.Anal.Bound.Elem. (2006) (Continuous casting of aluminium) I.Kovačević, B.Šarler, Materials Science and Engineering A (2006) (Phase field) G.Kosec, B.Šarler, Computer Modeling in Engineering and Sciences (2008) (Navier Stokes) G.Kosec, B.Šarler, J.Numer.Methods Heat & Fluid Flow (2008) (Porous media flow) R.Vertnik, B.Šarler, Cast Metals Research (2008) (Continuous casting of steel) G.Kosec, B.Šarler, Cast Metals Research (2008) (Freezing of pure metals) R.Vertnik, B.Šarler, Computer Modelig in Engineering and Sciences (2009) (Turbulent flow, k-epsilon model) Method improved and made available in solution of practical problems with the local formulation-LRBFCM. Solutions on several thousand nodes tested. Introduction
Solution of thermo-fluid problems with local approach –Local Radial Basis Function Collocation Method –Local pressure velocity coupling presented at FEF 2007 ∙Classical De Wahl Davis benchmark (Kosec, Šarler, 2008) (FEF2007) ∙Natural convection in porous media (Kosec, Šarler, 2008) ∙Meshless Approach to Solving Freezing with Natural Convection (Kosec, Šarler, in review) ∙Convection Driven Melting of Anisotropic Metals (Kosec, Šarler, in print) ∙Melting driven by natural convection (Kosec, Šarler, to be submitted) Extension of the method to the dynamic node configuration –Present work –Other’s research in the field (Libre, 2008, Bozzini, 2002, Gomez, 2006, Sarra, 2005, Schaback, 2000) Principal incitement –How to make LRBFCM h-adaptive in a most simple and effective way?
Local Radial Basis Functions Collocation Method Basis functions: Hardy’s multiquadrics LRBFCM RBF basis augmentation in order to improve the method, various functions might be included in the collocation basis. Constant and linear functions are added in the present work in order to enhance the method at the small sub- domain nodal distances regime. - Shape parameter - Spatial coordinate
The performance of the LRBFCM One-dimensional analysis is performed to asses the method’s behavior –The exponential function is taken as a test function Discretisation errors of the first and second derivatives are analyzed With respect to –The sub-domain nodal distances –The shape parameter C –Comparison of augmented to non-augmented LRBFCM - RBFCM representation - Exact values
Discretisation error of the derivatives. With respect to the nodal distance and shape parameter for non-augmented LRBFCM
Second derivative error. Comparison of non-augmented to the augmented LRBCM
The adaptation criteria An efficient criteria for adding and/or removing of the nodes is required –Derivatives calculation error extrapolation –Error predictor based on the shape of the field and/or properties of the numerical method The LRBFCM predictor –Based on the norm of LRBFCM collocation coefficients –No need for additional computation –Behaviour with respect to the spatial sub-domain configuration is similar to the behaviour of the derivatives error –Small number of calculation operations (fast) –Robust and very simple to implement Number of sub-domain nodes LRBCM coefficients
Error prediction. Comparison of error prediction to the actual derivative error
The adaptation algorithm rules Node insertion rules –If P is above the limit value Pmax the node is requested to expand (construct new nodes around it) –Level difference with it’s neighbors after the expansion is not allowed to exceed one –New node (child) can be constructed only when none of the neighbors has it’s child on the same coordinate –All child nodes are positioned symmetrical around parent node in order to retain equidistant sub-domain for all adaptation levels Node removal rules –If P is below the limit value Pmin the node is requested to close (remove child nodes) –Level difference with it’s neighbors after closing is not allowed to exceed one –Child node is allowed to be removed only if none of the neighbor parent node is pointing to it
The sub domain configuration Two types of the sub-domains appear (normal/rotated) Five nodded sub-domain The neighbor's neighbor is not necessarily the same node All sub-domains are equidistant
The refinement algorithm
Node expansion example Initial nodes distribution
Node expansion example Level one
Node expansion example Level two
Node expansion example Level three
Node expansion example Level four
Node expansion example Level five
Node expansion example Level six
First numerical example A two-dimensional time-dependent diffusion equation with Dirichlet boundary conditions All values are dimensionless Analytical solution
Test cases in the first numerical example Common settings Diffusivity is set to one Domain size is 1x1 Boundary conditions and initial state are the same for all cases All boundaries are set to zero Initial state is set to one Explicit time stepping with time step dt=1e-5 Case definitions Four different cases with different average nodes count through whole simulation are considered Each case with the adaptive node distribution (A) is compared with the case exhibiting the fixed node distribution (S) with the same average number of the nodes All values are dimensionless CaseAverage number of the nodes Case Case Case Case1-4961
First numerical example results Average error (left) and number of the nodes (right) with respect to time for cases 1-1 and 1-2.
First numerical example results Average error (left) and number of the nodes (right) with respect to time for cases 1-3 and 1-4.
Cross-section error comparison
Time evolution of the first numerical case
Second numerical example Convection dominated transport Neumann boundary conditions on all walls Adiabatic walls Initial state set to zero except on a small region where the initial state is one Circular velocity field High Peclet number (L – domain length) Results with velocity v after one revolution are compared with the results where the velocity is set to zero. The difference is introduced as a deviation Where and stands for simulation time (2π), temperature computed without advection term, temperature computed with the advection term.
Common settings Diffusivity is set to 2e-4 Domain size is 2x2 Boundary conditions and initial state are for all cases the same Explicit time stepping with time step dt=1e-5 Angular velocity is 1 and simulation time is 2π Case definitions Four different cases with different average nodes count through whole simulation Each case with adaptive node distribution (A) is compared to the case with static node distribution (S) with the same average nodes count. Test cases for the second numerical example CaseAverage number of the nodes Case Case Case Case All values are dimensionless
Second numerical example results – case 1
Second numerical example results – case 2
Second numerical example results – case 3
Second numerical example results – case 4
Time evolution of the second numerical case
Comparison – adaptive vs. static node distribution Adaptive nodes distribution Static nodes distribution
Conclusions The developed algorithm is ready to be implemented in more complex physical systems Fluid mechanics with iterative pressure velocity coupling Extension to the multi-phase systems (phase field, level set formulations) Melting and freezing Solidification of a binary system... Issues that need further clarification: –Behavior of the LRBFCM near the nodal density jumps –Rotated domain inclusion effects –Setting the free parameter –Sub-domain size influence –Sub-domain shape influence –Multi-grid behavior
Further development Implementation of the developed adaptive numerical approach with completely local pressure velocity coupling in more complex systems The local fluid flow computational approach has been extensively tested with good agreement with other authors There are some difficulties in implementing it with the developed adaptation algorithm
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Numerical error of the divative With respect to the sub-domain shape The LRBFCM achieves best performance with the equidistant sub domain (R=1)
Numerical error of the derivative With respect to curvature of the function