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Well-posed variational formulations of Friedrichs-type systems
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.ORCID-id: 0000-0003-0473-3263
Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
2021 (Engelska)Ingår i: Journal of Differential Equations, ISSN 0022-0396, E-ISSN 1090-2732, Vol. 292, s. 90-131Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

All finite element methods, as well as much of the Hilbert-space theory for partial differential equations, rely on variational formulations, that is, problems of the type: find u∈V such that a(v,u)=l(v) for each v∈L, where V,L are Sobolev spaces. However, for systems of Friedrichs type, there is a sharp disparity between established well-posedness theories, which are not variational, and the very successful discontinuous Galerkin methods that have been developed for such systems, which are variational. In an attempt to override this dichotomy, we present, through three specific examples of increasing complexity, well-posed variational formulations of boundary and initial–boundary-value problems of Friedrichs type. The variational forms we introduce are generalizations of those used for discontinuous Galerkin methods, in the sense that inhomogeneous boundary and initial conditions are enforced weakly through integrals in the variational forms. In the variational forms we introduce, the solution space is defined as a subspace V of the graph space associated with the differential operator in question, whereas the test function space L is a tuple of L2 spaces that separately enforce the equation, boundary conditions of characteristic type, and initial conditions.

Ort, förlag, år, upplaga, sidor
Elsevier, 2021. Vol. 292, s. 90-131
Nationell ämneskategori
Matematik
Identifikatorer
URN: urn:nbn:se:umu:diva-175317DOI: 10.1016/j.jde.2021.05.002ISI: 000656995900003Scopus ID: 2-s2.0-85105569339OAI: oai:DiVA.org:umu-175317DiVA, id: diva2:1470558
Forskningsfinansiär
Vetenskapsrådet, 2018-03546
Anmärkning

Previously included in thesis in manuscript form.

Tillgänglig från: 2020-09-25 Skapad: 2020-09-25 Senast uppdaterad: 2023-09-05Bibliografiskt granskad
Ingår i avhandling
1. The fW-mean filter framework for topology optimization and analysis of Friedrichs systems
Öppna denna publikation i ny flik eller fönster >>The fW-mean filter framework for topology optimization and analysis of Friedrichs systems
2020 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Alternativ titel[sv]
Ett ramverk för medelvärdesfilter inom topologioptimering samt analys av Friedrichssystem
Abstract [en]

Part I. Topology optimization is the most general form of design optimization in which the optimal layout of material within a given region of space is to be determined. Filters are essential components of many successful density based topology optimization approaches. The generalized fW-mean filter framework developed in this thesis provides a unified platform for construction, analysis, and implementation of filters. Extending existing algorithms, we demonstrate that under special albeit relevant conditions, the computational complexity of evaluating generalized fW-mean filters and their derivatives is linear in the number of design degrees of freedom. We prove that generalized fW-mean filters guarantee existence of solutions to the penalized minimum compliance problem, the archetypical problem in density based topology optimization. In this problem, the layout of linearly elastic material that minimizes the compliance given static supports and loads is to be determined. We formalize the connection between mathematical morphology and the notion of minimum length scale of a layout of material and thereby provide a theoretical foundation for imposing and assessing minimum length scales in density based topology optimization. Elaborating on the fact that some sequences of generalized fW-mean filters provide differentiable approximations of morphological operators, we devise a method capable of imposing different minimum length scales on the two material phases in minimum compliance problems.

 

Part II. The notion of Friedrichs systems, also known as symmetric positive systems, encompasses many linear models of physical phenomena. The prototype model is Maxwell's equations, which describe the evolution of the electromagnetic field in the presence of electrical charges and currents. In this thesis, we develop well-posed variational formulations of boundary and initial–boundary value problems of Friedrichs systems on bounded domains. In particular, we consider an inhomogeneous initial–boundary value problem that models lossless propagation of acoustic disturbances in a stagnant fluid. Galbrun's equation is a linear second order vector differential equation in the so-called Lagrangian displacement, which was derived to model lossless propagation of acoustic disturbances in the presence of a background flow. Our analysis of Galbrun's equation is centered on the observation that solutions to Galbrun's equation may be formally constructed from solutions to linearized Euler's equations. More precisely, the Lagrangian displacement is constructed as the solution to a transport-type equation driven by the Eulerian velocity perturbation. We present partial results on the well-posedness of Galbrun's equation in the particular case that the background flow is everywhere tangential to the domain boundary by demonstrating mild well-posedness of an initial–boundary value problem for linearized Euler's equations and that our construction of the Lagrangian displacement is well-defined. Moreover, we demonstrate that sufficiently regular solutions to Galbrun's equation satisfy an energy estimate.

Ort, förlag, år, upplaga, sidor
Umeå: Umeå universitet, Institutionen för datavetenskap, 2020. s. 49
Serie
Report / UMINF, ISSN 0348-0542 ; 20.09
Nyckelord
topology optimization, filters, mathematical morphology, size control, minimum compliance problem, Friedrichs systems, well-posedness, variational formulations, linearized Euler’s equations, Galbrun’s equation, acoustics
Nationell ämneskategori
Beräkningsmatematik
Forskningsämne
matematik
Identifikatorer
urn:nbn:se:umu:diva-175319 (URN)978-91-7855-368-6 (ISBN)978-91-7855-367-9 (ISBN)
Disputation
2020-10-22, Ma121, MIT-huset, Umeå universitet, Umeå, 14:00 (Engelska)
Opponent
Handledare
Tillgänglig från: 2020-10-01 Skapad: 2020-09-25 Senast uppdaterad: 2020-09-29Bibliografiskt granskad

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Berggren, MartinHägg, Linus

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