An Interface-Strip Domain Decomposition Preconditioner

Quarteroni, Alfio; Sala, Marzio; Valli, Alberto

doi:10.1137/04061057x

Cited by 3 publications

(4 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Mesh data structures are partitioned using the so-called vertex-oriented distribution [36], which is followed by most high-performance scientific packages like Trilinos [22], hypre [19] and PETSc [8]. As a result, ParFE can be easily interfaced with the aforementioned libraries.…”

Section: Software Issuesmentioning

confidence: 99%

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Arbenz

Lenthe

Mennel

et al. 2007

Numerical Meth Engineering

Self Cite

132

106

View full text Add to dashboard Cite

Abstract. The recent advances in microarchitectural bone imaging are disclosing the possibility to assess both the apparent density and the trabecular microstructure of intact bones in a single measurement. Coupling these imaging possibilities with microstructural finite element (µFE) analysis offers a powerful tool to improve bone stiffness and strength assessment for individual fracture risk prediction.Many elements are needed to accurately represent the intricate microarchitectural structure of bone; hence, the resulting µFE models possess a very large number of degrees of freedom. In order to be solved quickly and reliably on state-of-the-art parallel computers, the µFE analyses require advanced solution techniques. In this paper, we investigate the solution of the resulting systems of linear equations by the conjugate gradient algorithm, preconditioned by aggregation-based multigrid methods. We introduce a variant of the preconditioner that does not need assembling the system matrix but uses element-by-element techniques to build the multilevel hierarchy. The preconditioner exploits the voxel approach that is common in bone structure analysis, it has modest memory requirements, while being at the same time robust and scalable.Using the proposed methods, we have solved in less than 10 minutes a model of trabecular bone composed of 247'734'272 elements, leading to a matrix with 1'178'736'360 rows, using only 1024 CRAY XT3 processors. The ability to solve, for the first time, large biomedical problems with over 1 billion degrees of freedom on a routine basis will help us improve our understanding of the influence of densitometric, morphological and loading factors in the etiology of osteoporotic fractures such as commonly experienced at the hip, the spine and the wrist.

show abstract

Section: Software Issuesmentioning

confidence: 99%

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Arbenz

Lenthe

Mennel

et al. 2007

Numerical Meth Engineering

Self Cite

132

106

View full text Add to dashboard Cite

show abstract

“…The origin of this loss is the bound of the coarse component u 0 associated with the subdomain edges; see (20) and (22). Our experiments indicate that the bound of Theorem 1 is not sharp while those in [1] are.…”

Section: Instead Of (1+log(h/ ))(1+log(h/ H))mentioning

confidence: 68%

“…This result is weaker than the main result in [1] in that we have a factor (1+log(H/ h)) 2 instead of (1+log(H/ ))(1+log(H/ h)). The origin of this loss is the bound of the coarse component u 0 associated with the subdomain edges; see (20) and (22). Our experiments indicate that the bound of Theorem 1 is not sharp while those in [1] are.…”

Section: Remarkmentioning

confidence: 81%

See 1 more Smart Citation

Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity

Dohrmann

Widlund

2009

Numerical Meth Engineering

View full text Add to dashboard Cite

SUMMARYOverlapping Schwarz methods are considered for mixed finite element approximations of linear elasticity, with discontinuous pressure spaces, as well as for compressible elasticity approximated by standard conforming finite elements. The coarse components of the preconditioners are based on spaces, with a number of degrees of freedom per subdomain which are uniformly bounded, which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type, i.e. methods based on nonoverlapping decompositions of the domain. The local components of the new preconditioners are based on solvers on a set of overlapping subdomains.In the current study, the dimension of the coarse spaces is smaller than in recently developed algorithms; in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed. In many cases, this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered. In addition, in spite of using overlapping subdomains to define the local components of the preconditioner, values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid Schwarz algorithm. The use of discontinuous pressures makes it possible to work exclusively with symmetric, positive-definite problems and the standard preconditioned conjugate gradient method.Bounds are established for the condition number of the preconditioned operators. The bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains, and it is independent of the Poisson ratio as well as jumps in the Lamé parameters across the interface between the subdomains. Numerical results illustrate the findings.

show abstract

The research of Alberto Valli

Rodríguez¹,

Veiga²,

Quarteroni³

2016

Discrete &Amp; Continuous Dynamical Systems - S

Self Cite

View full text Add to dashboard Cite

The scientific activity of Professor Alberto Valli has been mainly devoted to three different subjects: theoretical analysis of partial differential equations in fluid dynamics; domain decomposition methods; numerical approximation of problems arising in low-frequency electromagnetism.The first problem he studied is the topic of his "tesi di laurea" (during the seventies this could be compared to a today's master thesis): the global-in-time existence of the solution to the two-dimensional Euler equations of incompressible inviscid fluids in a moving domain ([4]). This result, that can be seen as a first partial step towards to solution of the free-boundary problem, still an open problem in its full generality, is a non-straightforward extension of the classical existence result by Kato, and is a nice piece of work for a 22-year old researcher.The analysis of Euler equations continued under the guide and with the strong support of Hugo Beirão da Veiga in the papers [7], [8], [9], [10], [12]. The novelty was in the fact that the density of the fluid was not considered equal to a constant, but as an additional unknown of the problem (thus dealing for the first time with the analysis of non-homogeneous fluids, for which a new transport equation for the density has to be taken into account). Local-in-time existence was obtained by a suitable fixed point procedure.A further step of the research came when Alberto started to consider compressible fluids, this turn in the viscous case. The subject had been firstly faced by Matsumura and Nishida, but the analysis of qualitative properties as periodicity, steadiness and stability was still missing. The problem was solved in a series of papers ([16], [24], [25]), one of them in collaboration with Wojciech Zaj ' aczkowski: this last work is one of the most quoted papers of Alberto.Another result on compressible fluids deserves attention: the local-in-time existence of the solution of the free boundary problem for the complete nonlinear system describing the flow of viscous compressible fluids (see [15]). This is the only paper that Alberto and Paolo wrote together (with a very low approval on "Mathematical Reviews"...), and is recognized as one of the papers that have opened the understanding of this subject for compressible fluids.A paper on a different aspect of fluid dynamics is [20], devoted to the integral representation of the solution to the Stokes problem. The result comes from a direct and somehow standard use of Green functions and potential theory, and its main interest resides in the fact that the previous literature on this classical subject was quite imprecise. This paper highlights a recurrent feature of Alberto working attitude: the attention at the scientific framework of a problem and at the preceding contributions of other authors.At the end of the eighties Alberto shifted his interest towards numerical approximation of partial differential equations. The encounter with Alfio Quarteroni was an xi xii

show abstract

An Interface-Strip Domain Decomposition Preconditioner

Cited by 3 publications

References 30 publications

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

A scalable multi‐level preconditioner for matrix‐free µ‐finite element analysis of human bone structures

Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity

The research of Alberto Valli

Contact Info

Product

Resources

About