A semi‐local spectral/hp element solver for linear elasticity problems

Yu, Yue; Bittencourt, Marco L.; Karniadakis, George Em

doi:10.1002/nme.4739

Cited by 4 publications

(3 citation statements)

References 44 publications

(102 reference statements)

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“…With the local mass matrix (2.8) and the local stiffness matrix (2.9), one can then assemble the global mass matrix M and the global stiffness matrix K , respectively. For more details, we refer the interested readers to [36]. …”

Section: Mathematical Formulationmentioning

confidence: 99%

Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Perdikaris

Karniadakis

2016

Journal of Computational Physics

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We develop efficient numerical methods for fractional order PDEs, and employ them to investigate viscoelastic constitutive laws for arterial wall mechanics. Recent simulations using one-dimensional models [1] have indicated that fractional order models may offer a more powerful alternative for modeling the arterial wall response, exhibiting reduced sensitivity to parametric uncertainties compared with the integer-calculus-based models. Here, we study three-dimensional (3D) fractional PDEs that naturally model the continuous relaxation properties of soft tissue, and for the first time employ them to simulate flow structure interactions for patient-specific brain aneurysms. To deal with the high memory requirements and in order to accelerate the numerical evaluation of hereditary integrals, we employ a fast convolution method [2] that reduces the memory cost to O(log(N)) and the computational complexity to O(N log(N)). Furthermore, we combine the fast convolution with high-order backward differentiation to achieve third-order time integration accuracy. We confirm that in 3D viscoelastic simulations, the integer order models strongly depends on the relaxation parameters, while the fractional order models are less sensitive. As an application to long-time simulations in complex geometries, we also apply the method to modeling fluid–structure interaction of a 3D patient-specific compliant cerebral artery with an aneurysm. Taken together, our findings demonstrate that fractional calculus can be employed effectively in modeling complex behavior of materials in realistic 3D time-dependent problems if properly designed efficient algorithms are employed to overcome the extra memory requirements and computational complexity associated with the non-local character of fractional derivatives.

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Section: Mathematical Formulationmentioning

confidence: 99%

Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Perdikaris

Karniadakis

2016

Journal of Computational Physics

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“…16 On the other hand, in partitioned FSI approaches, the governing system of equations is split into separate fluid and structure solves, thus promoting software modularity and scalability to large-scale parallel computing environments. [19][20][21] Moreover, numerical stability issues can be overcome by strategies based on fixed-point iterations and Aitken relaxation, leading to stable coupling schemes even for small mass ratios -a common problem in FSI simulations. 19,20 Simulations presented in this work have been performed using our research spectral/hp-element solver NekTar.…”

Section: Man -Computational Aspectsmentioning

confidence: 99%

“…19,20 Simulations presented in this work have been performed using our research spectral/hp-element solver NekTar. 13,[19][20][21][22][23][24] For a detailed exposition to the numerical methods and implementational aspects of this work, the reader is referred to the bibliography. Finally, we note that one way to scale macro-scale methods to extremely large problems results by applying the multi-patch domain decomposition introduced by Grinberg and Karniadakis, 25 where a large domain is divided into a series of loosely coupled overlapping subdomains (patches) of manageable size.…”

Section: Man -Computational Aspectsmentioning

confidence: 99%

Multiscale modeling and simulation of brain blood flow

2016

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The aim of this work is to present an overview of recent advances in multi-scale modeling of brain blood flow. In particular, we present some approaches that enable the study of multi-scale and multi-physics phenomena in the cerebral vasculature. We discuss the formulation of continuum and atomistic modeling approaches, present a consistent framework for their concurrent coupling, and list some of the challenges that one needs to overcome in achieving a seamless and scalable integration of heterogeneous numerical solvers. The effectiveness of the proposed framework is demonstrated in a realistic case involving modeling the thrombus formation process taking place on the wall of a patient-specific cerebral aneurysm. This highlights the ability of multi-scale algorithms to resolve important biophysical processes that span several spatial and temporal scales, potentially yielding new insight into the key aspects of brain blood flow in health and disease. Finally, we discuss open questions in multi-scale modeling and emerging topics of future research.

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Spectral Element andhpMethods

Kirby

Karniadakis

2017

Encyclopedia of Computational Mechanics Second Edition

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Spectral/ hp element methods provide high‐order discretization, which is essential in the longtime integration of advection–diffusion systems and for capturing dynamic instabilities in solids. In this chapter, we review the main formulations for simulations of incompressible and compressible viscous flows as well as for solid mechanics and present several examples with some emphasis on fluid–structure interactions and interfaces. The first generation of (nodal) spectral elements was limited to relatively simple geometries and smooth solutions. However, the new generation of spectral/ hp elements, consisting of both nodal and modal forms, can handle very complex geometries using unstructured grids and can capture strong shocks by employing discontinuous Galerkin methods. New flexible formulations allow simulations of multiphysics problems including extremely complex geometries and multiphase flows. Several implementation strategies have also been developed on the basis of multilevel parallel algorithms that allow dynamic ‐refinement at constant wall clock time. After three decades of intense developments, spectral element and hp methods are mature and efficient to be used effectively in applications of industrial complexity. They provide the capabilities that standard finite element and finite volume methods do, but, in addition, they exhibit high‐order accuracy and error control.

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A semi‐local spectral/hp element solver for linear elasticity problems

Cited by 4 publications

References 44 publications

Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Fractional modeling of viscoelasticity in 3D cerebral arteries and aneurysms

Multiscale modeling and simulation of brain blood flow

Spectral Element andhpMethods

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