| 1. |
- Nguyen, Van Dang, et al.
(författare)
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A partition of unity finite element method for computational diffusion MRI
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- The Bloch-Torrey equation describes the evolution of the spin (usually water proton) magnetization under the influence of applied magnetic field gradients and is commonly used in numerical simulations for diffusion MRI and NMR. Microscopic heterogeneity inside the imaging voxel is modeled by interfaces inside the simulation domain, where a discontinuity in the magnetization across the interfaces is produced via a permeability coefficient on the interfaces. To avoid having to simulate on a computational domain that is the size of an entire imaging voxel, which is often much larger than the scale of the microscopic heterogeneity as well as the mean spin diffusion displacement, smaller representative volumes of the imaging medium can be used as the simulation domain. In this case, the exterior boundaries of a representative volume either must be far away from the initial positions of the spins or suitable boundary conditions must be found to allow the movement of spins across these exterior boundaries. Many efforts have been made to solve the equation but there is still missing an efficient high performance computing framework. In this work, we present formulations of the interface as well as the exterior boundary conditions that are computationally efficient and suitable for arbitrary order finite elements and parallelization. In particular, the formulations use extended finite elements with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as operator splitting for the exterior boundary conditions. The method appears to be straightforward to implement and it is implemented in the FEniCS for moderate-scale simulations and in the FEniCS-HPC for the large-scale simulations. The accuracy of the resulting method is validated numerically and a good scalability is shown for the parallel implementation. We show that the simulated dMRI signals offer good approximations to reference signals in cases where the latter are available. Finally, we do simulations on a complex neuron to study how the signals decay under the effect of the permeable membrane and to show that the method can be used to simulate for complex geometries that we have not done before.Highlights:The discontinuity in the magnetization across the interior interfaces of the medium is weakly imposed, allowing generalization to arbitrary order finite elements.Spin exchange across the external boundaries is implemented by weakly imposing an artificial, high permeability, condition, allowing generalization to non-matching meshes.Thus, optimal convergence with respect to the space discretization is achieved.The second-order Crank-Nicolson method is chosen for the time discretization to reduce oscillations at high gradient strengths and allows for larger time-step sizes.The method is of a high level of simplicity and suitable for parallelization.An efficient open-source code is implemented in the FEniCS and FEniCS-HPC platforms.
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| 2. |
- Jansson, Johan, et al.
(författare)
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Adaptive simulation of unsteady flow past the submerged part of a floating wind turbine platform
- 2015
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Ingår i: MARINE 2015 - Computational Methods in Marine Engineering VI. - : International Center for Numerical Methods in Engineering (CIMNE). - 9788494392863 ; , s. 35-46
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Konferensbidrag (refereegranskat)abstract
- Offshore floating platforms for wind turbines represent challenging concepts for designers trying to combine an optimal compromise between cost effectiveness and performance. Modelling of the hydrodynamic behaviour of the structure is still the subject of wide debate in the technical communities. The assessment of the hydrodynamics of the support structure is not an easy task as the floaters consist of an assembly of columns, braces and pontoons, commonly also with heave plates: Each of these components corresponds to a different hydrodynamic model and it further interacts with the other elements. This results in very complex non-linear modeling, which makes it necessary to resort to computational fluid dynamics (CFD) methods for the evaluation of the combined hydrodynamics. In the framework of the collaboration between the Basque Centre for Applied Mathematics (BCAM) and Tecnalia R&I, the interaction of the sea flow with a semisubmersible floating offshore wind platform have been calculated by using the open source solver Unicorn in the FEniCS-HPC framework when subject to a steady inflow. The prototype of the platform consists in a semi-submersible 4-columns column stabilized platform - NAUTILUS Floating Solutions concept-; columns are connected by a rigid ring pontoon provided with heave damping plates at the bottom. The novelty of the approach in FEniCS-HPC hinges upon an implicit formulation for the turbulence, a cheap free slip model of the boundary layer and goal-oriented mesh adaptivity [8, 6, 9, 20, 1]. We find that the results are consistent with experimental results for cylinders at high Reynolds number.
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| 3. |
- Hoffman, Johan, 1974-, et al.
(författare)
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DOLFIN: Dynamic Object oriented Library for FINite element computation
- 2002
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this report, we present a new software project intended to provide a platform for daptive finite element computation in both research and education. The software DOLFIN is implemented in C++ and provides an object-oriented powerful and flexible environment in which finnite element methods for differential equations can be easily implemented. The current version of DOLFIN already includes a nnumber of modules for special differential equations, including the incompressible Navier-Stokes equations. DOLFIN is free and open source, which we believe is a key point in creating truly useful software.
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| 4. |
- Hoffman, Johan, 1974-, et al.
(författare)
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The FEniCS project
- 2003
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- The goal of the FEniCS project is to develop open-source software for the automation of Computational Mathematical Modeling (CMM). FEniCS is a joint project between the Toyota Technological Institute at Chicago, the University of Chicago, and Chalmers University of Technology. The vision of FEniCS is to set a new standard in CMM, which can be described as the Automation of CMM, towards the goals of generality, efficiency, and simplicity, concerning mathematical methodology, implementation, and application.
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| 5. |
- Jansson, Johan, 1978-, et al.
(författare)
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Automated error control in finite element methods withapplications in fluid flow
- 2014
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we present a new adaptive finite element method for thesolution of linear and non-linear partial differential equationsdirectly using the a posteriori error representation as a local errorindicator, with the primal and dual solutions approximated in the samefinite element space, here piecewise continuous linear functions onthe same mesh. Since this approach gives a global a posteriori errorrepresentation that is zero due to Galerkin orthogonality, the errorrepresentation has traditionally been thought to contain noinformation about the error. However, for elliptic andconvection-diffusion model problems we show the opposite, that locallythe orthogonal error representation behaves very similar to thenon-orthogonal error representation using a higher order approximationof the dual. We have previously proved an a priori estimate of thelocal error indicator for elliptic problems, and in this paper weextend the proof to convection-reaction problems. We also present aversion of the method for non-elliptic and non-linear problems using astabilized finite element method where the a posteriori errorrepresentation is no longer orthogonal. We apply this method to thestationary incompressible Navier-Stokes equation and perform detailednumerical experiments which show that the a posteriori error estimateis within a factor 2 of the error based on a reference value on a finemesh, except in a few data points on very coarse meshes for anon-smooth test case where it is within a factor 3.
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| 6. |
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| 7. |
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| 8. |
- Hoffman, Johan, et al.
(författare)
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FEniCS-HPC: Coupled Multiphysics in Computational Fluid Dynamics
- 2017
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Ingår i: High-Performance Scientific Computing. - Cham : Springer. - 9783319538617 - 9783319538624 ; , s. 58-69
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Konferensbidrag (refereegranskat)abstract
- We present a framework for coupled multiphysics in computational fluid dynamics, targeting massively parallel systems. Our strategy is based on general problem formulations in the form of partial differential equations and the finite element method, which open for automation, and optimization of a set of fundamental algorithms. We describe these algorithms, including finite element matrix assembly, adaptive mesh refinement and mesh smoothing; and multiphysics coupling methodologies such as unified continuum fluid-structure interaction (FSI), and aeroacoustics by coupled acoustic analogies. The framework is implemented as FEniCS open source software components, optimized for massively parallel computing. Examples of applications are presented, including simulation of aeroacoustic noise generated by an airplane landing gear, simulation of the blood flow in the human heart, and simulation of the human voice organ.
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| 9. |
- Nguyen, Van Dang, 1985-, et al.
(författare)
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A partition of unity finite element method for computational diffusion MRI
- 2018
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Ingår i: Journal of Computational Physics. - : Elsevier. - 0021-9991 .- 1090-2716. ; 375, s. 271-290
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Tidskriftsartikel (refereegranskat)abstract
- The Bloch–Torrey equation describes the evolution of the spin (usually water proton) magnetization under the influence of applied magnetic field gradients and is commonly used in numerical simulations for diffusion MRI and NMR. Microscopic heterogeneity inside the imaging voxel is modeled by interfaces inside the simulation domain, where a discontinuity in the magnetization across the interfaces is produced via a permeability coefficient on the interfaces. To avoid having to simulate on a computational domain that is the size of an entire imaging voxel, which is often much larger than the scale of the microscopic heterogeneity as well as the mean spin diffusion displacement, smaller representative volumes of the imaging medium can be used as the simulation domain. In this case, the exterior boundaries of a representative volume either must be far away from the initial positions of the spins or suitable boundary conditions must be found to allow the movement of spins across these exterior boundaries.Many approaches have been taken to solve the Bloch–Torrey equation but an efficient high-performance computing framework is still missing. In this paper, we present formulations of the interface as well as the exterior boundary conditions that are computationally efficient and suitable for arbitrary order finite elements and parallelization. In particular, the formulations are based on the partition of unity concept which allows for a discontinuous solution across interfaces conforming with the mesh with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as an operator splitting for the exterior boundary conditions. The method is straightforward to implement and it is available in FEniCS for moderate-scale simulations and in FEniCS-HPC for large-scale simulations. The order of accuracy of the resulting method is validated in numerical tests and a good scalability is shown for the parallel implementation. We show that the simulated dMRI signals offer good approximations to reference signals in cases where the latter are available and we performed simulations for a realistic model of a neuron to show that the method can be used for complex geometries.
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| 10. |
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