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Multiscale Methods for Wave Propagation Problems
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Simulations of wave propagation in heterogeneous media and at high frequencies are important in many applications such as seismic-, {electro-magnetic-,} acoustic-, fluid flow problems and others. These are classical multiscale problems and often too computationally expensive for direct numerical simulation. The smallest scales must be well resolved over a computational domain represented by the largest scale and this results in a very high computational cost. We develop and analyze numerical techniques based on the heterogeneous multiscale method (HMM) framework for such wave equations with highly oscillatory solutions $u^{\varepsilon}$ where $\varepsilon$ represents the size of the smallest scale. In these techniques the oscillatory microscale is approximated on small local microproblems of size $\varepsilon$ in spatial and time directions. The solution of the microproblems are then coupled to a global macroscale model in divergence form $u_{tt} = \nabla \cdot F$ where the flux $F$ is obtained from the microproblems. The oscillations can either originate from fluctuations in the velocity coefficients or from high frequency initial and boundary conditions. We have developed algorithms that couple micro and macroscales for both these cases. The choice of macroscale variables is inspired by the analytic theories of homogenization and geometrical optics respectively. In the first case local averages $u \approx u^{\varepsilon}$ are used on the macroscale. In the second case, phase $\phi$ and energy are natural macroscopic variables. There are two major goals of this research. One goal is to develop and analyze algorithms for simulating multiscale wave propagation with low computational complexity, and even independent of $\varepsilon$ for finite time problems. This is seen in many examples in one, two and three dimensions. The other goal is to use wave propagation as a model to better understand the HMM framework. An example in this direction is simulation with oscillatory wave field over long time. The dispersive effects that then occur is well approximated by a HMM method that was originally formulated for finite time where added accuracy is required but no explicit adjustment to include dispersion, an evidence of the robustness of the method.

Abstract [sv]

Simulering av högfrekventa vågor i heterogena material är viktigt i många tillämpningar, till exempel seismologi, elektromagnetism, akustik och  strömningsmekanik. Dessa tillämpningar är exempel på klassiska multiskalproblem och har typiskt en för hög beräkningskostnad, i form av datortid och minne, för en direkt numerisk simulering. De minsta skalorna i problemet måste vara upplösta över ett område som representeras av dom största skalorna och detta innebär en hög beräkningskostnad. Vi har utvecklat och analyserat numeriska metoder för vågekvationer med snabbt oscillerande lösningar $u^{\varepsilon}$ där $\varepsilon$ representerar storleken på den minsta skalan. Metoderna är baserade på ramverket \emph{heterogena multiskalmetoden} (HMM). I dessa metoder approximeras den hastigt oscillerande mikroskalan med små lokala mikroproblem av storleksordning $\varepsilon$ i tids- och rumsriktning. Lösningen till mikroproblemen är kopplade till en global modell på makroskalan i divergensform $u_{tt} = \nabla \cdot F$, där flödet $F$ ges av mikroproblemen. De hastiga oscillationerna kan härröras från snabba variationer i hastighetsfältet, begynnelsevillkor eller randvillkor. Vi har utvecklat algoritmer som kopplar mikro- och makroskalor i bägge fallen. Valet av makroskalvariabler inspireras av de analytiska metoderna homogenisering och geometrisk optik. I det första fallet används lokala medelvärden $u \approx u^{\varepsilon}$ på makroskalnivån. I det andra fallet är fas $\phi$ och energi bra val av makroskalvariabler. Det finns två huvudmål med vår forskning. Ett mål är att utveckla och analysera algoritmer för simulering av vågproblem med multipla skalor med låg beräkningskostnad (om möjligt, oberoende av $\varepsilon$) för problem över begränsad tid. Vi visar numeriska resultat från multiskalproblem i en, två och tre dimensioner. Det andra målet är att att använda vågutbredning som en modell för att bättre förstå HMM ramverket. Ett exempel på detta är simulering med oscillerande hastighetsfält över lång tid. Efter lång tid så uppträder dispersion. Vi har demonstrerat att vår HMM-metod, som ursprungligen var formulerad för begränsad tid, även kan appliceras på detta fall. För att få den rätta dispersionen krävs högre noggrannhetsordning, men metoden ändrar inte form. Detta visar på metodens robusthet.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology , 2011. , xi, 79 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2011:16
National Category
Computational Mathematics
Identifiers
ISBN: 978-91-7501-176-9OAI: oai:DiVA.org:kth-48072DiVA: diva2:456767
Public defence
2011-12-09, E3, Lindstedtsvägen 3, Stockholm, 10:00 (English)
Funder
Swedish e‐Science Research Center
Note
QC 20111117Available from: 2011-11-17 Created: 2011-11-15 Last updated: 2012-05-24Bibliographically approved
List of papers
1. Multiscale methods for the wave equation
Open this publication in new window or tab >>Multiscale methods for the wave equation
2007 (English)In: PAMM · Proc. Appl. Math. Mech. 7, 2007, 1140903-1140904 p.Conference paper (Other academic)
Abstract [en]

We consider the wave equation in a medium with a rapidly varying speed of propagation. We construct a multiscale schemebased on the heterogeneous multiscale method, which can compute the correct coarse behavior of wave pulses traveling in themedium, at a computational cost essentially independent of the size of the small scale variations. This is verified by theoreticalresults and numerical examples.

Keyword
multiscale methods, HMM, wave equation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-48055 (URN)10.1002/pamm.200700930 (DOI)
Conference
ICIAM07 / Wiley InterScience
Note
QC 20111117Available from: 2011-11-17 Created: 2011-11-15 Last updated: 2012-02-03Bibliographically approved
2. Multi-scale methods for wave propagation in heterogeneous media
Open this publication in new window or tab >>Multi-scale methods for wave propagation in heterogeneous media
2011 (English)In: Communications in Mathematical Sciences, ISSN 1539-6746, E-ISSN 1945-0796, Vol. 9, no 1, 33-56 p.Article in journal (Refereed) Published
Abstract [en]

Multi-scale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multi-scale wave propagation in the framework of heterogeneous multi-scale method. The numerical methods couple simulations on macro-and micro-scales for problems with rapidly oscillating coefficients. We show that the complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the micro-scale. A convergence proof is given and numerical results are presented for periodic problems in one, two, and three dimensions. The method is also successfully applied to non-periodic problems and for long time integration where dispersive effects occur.

Keyword
Multi-scale, wave propagation, HMM, homogenization
Mathematics
Identifiers
urn:nbn:se:kth:diva-27092 (URN)000283985900002 ()2-s2.0-78049260965 (ScopusID)
Funder
Swedish e‐Science Research Center
Note
QC 20101209Available from: 2010-12-09 Created: 2010-12-06 Last updated: 2012-05-23Bibliographically approved
3. Multiscale Methods for Wave Propagation in Heterogeneous Media Over Long Time
Open this publication in new window or tab >>Multiscale Methods for Wave Propagation in Heterogeneous Media Over Long Time
2012 (English)In: Numerical Analysis of Multiscale Computations / [ed] Björn Engquist, Olof Runborg, Yen-Hsi R. Tsai, Springer Verlag , 2012, 167-186 p.Chapter in book (Other academic)
Abstract [en]

Multiscale wave propagation problems are computationally costly to solve by traditional techniques because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multiscale wave propagation in the framework of the heterogeneous multiscale method (HMM). The numerical methods couple simulations on macro- and microscales for problems with rapidly oscillating coefficients. The complexity of the new method is significantly lower than that of traditional techniques with a computational cost that is essentially independent of the smallest scale, when computing solutions at a fixed time and accuracy. We show numerical examples of the HMM applied to long time integration of wave propagation problems in both periodic and non-periodic medium. In both cases our HMM accurately captures the dispersive effects that occur. We also give a stability proof for the HMM, when it is applied to long time wave propagation problems.

Place, publisher, year, edition, pages
Springer Verlag, 2012
Series
, Lecture Notes in Computational Science and Engineering, ISSN 1439-7358
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-48058 (URN)10.1007/978-3-642-21943-6_8 (DOI)978-3-642-21942-9 (ISBN)
Funder
Swedish e‐Science Research Center
Note

QC 20111116

Available from: 2011-11-16 Created: 2011-11-15 Last updated: 2013-04-09Bibliographically approved
4. Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time
Open this publication in new window or tab >>Analysis of HMM for One Dimensional Wave Propagation Problems Over Long Time
2011 (English)Article in journal (Refereed) Submitted
Abstract [en]

Multiscale problems are computationally costly to solve by direct simulation because the smallest scales must be represented over a domain determined by the largest scales of the problem. We have developed and analyzed new numerical methods for multiscale wave propagation following the framework of the heterogeneous multiscale method. The numerical methods couple simulations on macro- and microscales for problems with rapidly fluctuating material coefficients. The computational complexity of the new method is significantly lower than that of traditional techniques. We focus on HMM approximation applied to long time integration of one-dimensional wave propagation problems in both periodic and non-periodic medium and show that the dispersive effect that appear after long time is fully captured.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-48311 (URN)
Note

QC 20111117

Available from: 2011-11-17 Created: 2011-11-17 Last updated: 2015-09-23Bibliographically approved
5. Multiscale Methods for One Dimensional Wave Propagation with High Frequency Initial Data
Open this publication in new window or tab >>Multiscale Methods for One Dimensional Wave Propagation with High Frequency Initial Data
Abstract [en]

High frequency wave propagation problems are computationally costly to solve by traditional techniques because the short wavelength must be well represented over a domain determined by the largest scales of the problem. We have developed and analyzed a new numerical method for high frequency wave propagation in the framework of heterogeneous multiscale methods, closely related to the analytical method of geometrical optics. The numerical method couples simulations on macro- and micro-scales for problems with highly oscillatory initial data. The method has a computational complexity essentially independent of the wavelength. We give one numerical example with a sharp but regular jump in velocity on the microscopic scale for which geometrical optics fails but our HMM gives correct results. We briefly discuss how the method can be extended to higher dimensional problems.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2011. 24 p.
Series
Trita-NA, ISSN 0348-2952 ; 2011:6
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-48068 (URN)
Note
QC 20111117Available from: 2011-11-17 Created: 2011-11-15 Last updated: 2011-11-17Bibliographically approved
6. Algorithms and Codes for Wave Propagation Problems
Open this publication in new window or tab >>Algorithms and Codes for Wave Propagation Problems
Abstract [en]

This technical report is a summary of selected numerical methods formultiscale wave propagation problems. The main topic is the discussionof nite dierence schemes, kernels for computing the mean value of oscil-latory functions and how to compute coecients in an eective equationfor long time wave propagation.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2011. 21 p.
Series
Trita-NA, ISSN 0348-2952 ; 2011:6
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-48067 (URN)
Note
QC 20111117Available from: 2011-11-17 Created: 2011-11-15 Last updated: 2011-11-17Bibliographically approved

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