Arbeiter, F.

Klimenko, D.

Pasler, M. J..

Piccioni Koch, D.

Prothmann, N.

Schlindwein, G.

Schulz, M. J.

Weth, Axel von der

Zinn, K.

English

KITopen

IKIT/INR/MET  (Maschinenbau)www.kit.eduE-mail: axel.vonderWeth@kit.edu1Numerical Solution Strategies in Permeation ProcessesAxel von der Weth, D. Piccioni Koch, F. Arbeiter, D. Klimenko, G. Schlindwein, M.  J. Schulz, V. Pasler, N. Prothmann K. Zinn29. 6. 2020, INRContents:1. Retrospection: Situation at gas release experiment2. Countermeasure: Algorithm using inverse elements3. One‐dimensional solvers4. Matrix inversion5. Results6. Current output= DSL 2021 , E-mail: axel.vonderWeth@kit.edu1.: Retrospection: Gas releaseexperiment GRID: Status ofDSL2019 and ICTT2019CAD study ofgas release setup(main parts)heatableenclosure(containingspecimen)mainvalvepressuregaugeenclosurepressure, concentrationin solid load phaseTimerelease phaseevac.Private communicationIdea for numerical solution of solving algorithm of PDE of transport type:U. v. Troissont DSL 2019	 	 		K. Nagato 		 		R. Dagan	 	12 	 	12 	 	 		 	 ,		 	 		12	 	 	̿	 ,	 	 	 	 	 		analytic numericAgreement with steady stateno yes Rediffusion no possible Under construction2DSL 2021 , E-mail: axel.vonderWeth@kit.edu2.: Countermeasure: Algorithms using inverse elements.	 	 	 ∗ , λ 0.01	, 0 1explicit Euler forward algorithm: 		 	 1 	k time index,dt time integration intervallimplicit Euler backward algorithm: 	 	 	 	 	 	11	 	 	 		 1 	11 	 	 		 	 	 	 	combined solver:Promising result: Sameaccuracy as other but withcombined solver a factor of 100increased time step3DSL 2021 , E-mail: axel.vonderWeth@kit.edu3.: One-dimensional solvers≈  	∆ ,	 2	 , , ==  	 	∗, 	 2	 , ,3.: One-dimensional solvers	 	 	 	 		 			1 ∗ 	, 	 , ∗	 , 	 2	 , ,			11 ∆1 2∆1 1 ∆0Cylindrical coordinates see presentation by T. Glage4DSL 2021 , E-mail: axel.vonderWeth@kit.edu4. Matrix Inversion	 ̿	 0	 ↔ 	 ̿	 ,	 	 	 	 	 	 	 	A worse result is better than nothing5Disadvantage: Possibly advantage of tridiagonal matrices lost! DSL 2021 , E-mail: axel.vonderWeth@kit.edu4.: Matrix inversion (B&B)b 1,1 1b , 1b 1, 0	, 1b , 0	, 1b ,	 	∗				1Initial valuesNote: B&B algorithm is used for two applications: Firstly way back from a measured permeation graph to Deffand ks, sa and determing the inverse matrix., 	1		∀	 	 	 	6DSL 2021 , E-mail: axel.vonderWeth@kit.edu7DSL 2021 , E-mail: axel.vonderWeth@kit.edu8Possibilties of speeding up B&B matrix inversion:̿ ̿DSL 2021 , E-mail: axel.vonderWeth@kit.edu9SSEV sufficiently mapped by B&B matrix inversion generated Euler backward solverDSL 2021 , E-mail: axel.vonderWeth@kit.edu5. Results:10- Easy programmable- Possibly positive and negative eigenvalues (D*=0.25)- Numerical instable case D*>0.5DSL 2021 , E-mail: axel.vonderWeth@kit.edu‐ Concentration vector using forward solver shows optimum at D*=0.25 (only positive eigenvalues) independent from number of elements compared with Daynes analytical solution [1] with 200 summands (10‐15)11[1] e. g. Crank, The Mathematics of Diffusion, eq. 4.24aDSL 2021 , E-mail: axel.vonderWeth@kit.edu- Backward solver shows a different behaviour of increasing error with increasing D*, generates disadvantage by differential quotient estimation, no optimum observed, especially observing eigenvalue spectra all positive infinite D* values possibleSee also presentation by T. Glage for analytical solution12DSL 2021 , E-mail: axel.vonderWeth@kit.edu1311 2	 ∗ 1 1	 ,1 0, 1 	 	 2	, 1,	 1DSL 2021 , E-mail: axel.vonderWeth@kit.edu‐ Constant error of Crank Nicolson solver observed, eigenvalue spectra agree only for D*<=0.25‐ Approximation of inverse matrix “huge” effort, see talk by T. Glage‐ All eigenvalues between ‐1 and +114DSL 2021 , E-mail: axel.vonderWeth@kit.edu Nearly constant error of the combined solver  ∗	 0.8	 Analytical eigenvalues not available Numerical eigenvalues seem to be fitting  ∗ ⁄ , see talk by T. Glage15DSL 2021 , E-mail: axel.vonderWeth@kit.edu6. Current output • D*=0.25 Euler forward is integrated into Q-PETEanalysis program• Computation time of first results (n=100) about 17 hours @inrclu, about 6 hours @ bwUniCluster2.0 and 40s  @ bwUniCluster2.0 using the D*=0.25 Euler forward solver generating “new” code structure• Time of calculation depends on the degree of non linearity (Re-Diffusion), up to 120 s @ bwUniCluster2.0 . • For B&B solving inverse problem getting from a measured curve D* is constant, D (ks, sa) comes from B&B and dt is adjusted!• First Arrhenius plot of Q-PETE results GRID is work in progress 16DSL 2021 , E-mail: axel.vonderWeth@kit.edu17Solver type Programmability Stability(D*) Eigenvalue spectrumaccuracyForward Euler easy 0.5 Critical, stable only D*<=0.25Very goodBackward Euler strong No limit Only positive poorCombined Solver strong 0.8 critical goodCrank Nicolson difficult No limit stable goodOstensive recommendationDSL 2021 , E-mail: axel.vonderWeth@kit.eduAcknowledgments: The authors acknowledge support by the state of Baden-Württemberg through bwHPC.Thank You for paying attention18

Numerical Solution Strategies in Permeation Processes

https://publikationen.bibliothek.kit.edu/1000134600/119340887

Numerical Solution Strategies in Permeation Processes

Abstract

Similar works

Full text

Available Versions

KITopen