| 1. |
- Alnæs, Martin S., et al.
(författare)
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The FEniCS Project Version 1.5
- 2015
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Ingår i: Archive of Numerical Software. - 2197-8263 .- 2197-8263. ; 3:100, s. 9-23
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Tidskriftsartikel (refereegranskat)abstract
- The FEniCS Project is a collaborative project for the development of innovative concepts and tools for automated scientific computing, with a particular focus on the solution of differential equations by finite element methods. The FEniCS Projects software consists of a collection of interoperable software components, including DOLFIN, FFC, FIAT, Instant, UFC, UFL, and mshr. This note describes the new features and changes introduced in the release of FEniCS version 1.5.
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| 2. |
- Alnaes, Martin, et al.
(författare)
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Unified Form Language: A Domain-Specific Language for Weak Formulations of Partial Differential Equations
- 2014
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Ingår i: ACM Transactions on Mathematical Software. - 0098-3500. ; 40:2, s. artikel nr 9-
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Tidskriftsartikel (refereegranskat)abstract
- We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and functionals, automatic differentiation of forms and expressions, arbitrary function space hierarchies for multifield problems, general differential operators and flexible tensor algebra. With these features, UFL has been used to effortlessly express finite element methods for complex systems of partial differential equations in near-mathematical notation, resulting in compact, intuitive and readable programs. We present in this work the language and its construction. An implementation of UFL is freely available as an open-source software library. The library generates abstract syntax tree representations of variational problems, which are used by other software libraries to generate concrete low-level implementations. Some application examples are presented and libraries that support UFL are highlighted.
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| 3. |
- Balaban, Gabriel, et al.
(författare)
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A Newton Method for Fluid-Structure Interaction Using Full Jacobians Based on Automatic Form Differentiation
- 2012
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Ingår i: 6th European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2012; Vienna; Austria; 10 September 2012 through 14 September 2012. - 9783950353709 ; , s. 3434-3447
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Konferensbidrag (refereegranskat)abstract
- The study of fluid-structure interaction (FSI) problems is becoming increasingly important both as part of design/engineering and in the modeling of biomedical processes. Examples include the design of new fighter aircraft, the study of the dynamics of heart valves, and the design of prosthetic heart valves. FSI problems are highly coupled and highly nonlinear problems which are challenging to solve. Furthermore, the solution of the discretized system of nonlinear equations is particularly challenging in cases where the solid and the fluid have densities of similar size; this is typically the case for the simulation of biomedical processes involving the deformation of tissue. In such cases, a simple fixed point iteration, in which the solution from a fluid solver is used to impose Neumann boundary conditions for a structure (elasticity), followed by an update of the fluid domain based on the structure solution (via the solution of an auxiliary problem for the update of the fluid mesh), may fail to converge. Instead, a more coupled approach such as a Newton or quasi-Newton method must be employed. In this note, we study the use of Newton's method to solve the fully coupled FSI problem. Typically, a Lagrangian formulation is used to describe the solid; that is, the solid equations are solved on a fixed reference domain (the initial configuration), while an ALE (Arbitrary Lagrangian-Eulerian) formulation is used to describe the fluid. This means that the fluid domain is changing throughout the simulation of a time-dependent problem. The differentiation of the FSI problem, which is required to formulate Newton's method, therefore involves a differentiation with respect to the changing domain of the fluid problem. Such shape differentiation can indeed be used to derive the full Jacobian of the FSI problem; see Fernández and Moubachir [3]. We here study an alternative approach based on mapping the fluid problem back to the initial configuration of the fluid domain. This alternative is advantageous since it allows the use of straightforward differentiation on a fixed domain. This also allows the use of existing tools for automatic differentiation of finite element variational forms such as those developed as part of the FEniCS Project [5-7]. The FEniCS form language UFL [1] is a domain-specific language for finite element variational forms which allows the FSI problem to be expressed in a language close to the mathematical notation. Forms may be differentiated automatically, and automatically assembled into matrices and vectors. The methodology is here applied to the fully nonlinear time-dependent FSI problem modeled by the incompressible Navier-Stokes equations and the St. Venant-Kirchoff nonlinear hyperelastic model.
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| 4. |
- Benedusi, Pietro, et al.
(författare)
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Modeling Excitable Cells with the EMI Equations : Spectral Analysis and Iterative Solution Strategy
- 2024
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Ingår i: Journal of Scientific Computing. - : Springer. - 0885-7474 .- 1573-7691. ; 98:3
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Tidskriftsartikel (refereegranskat)abstract
- In this work, we are interested in solving large linear systems stemming from the extra-membrane-intra model, which is employed for simulating excitable tissues at a cellular scale. After setting the related systems of partial differential equations equipped with proper boundary conditions, we provide its finite element discretization and focus on the resulting large linear systems. We first give a relatively complete spectral analysis using tools from the theory of Generalized Locally Toeplitz matrix sequences. The obtained spectral information is used for designing appropriate preconditioned Krylov solvers. Through numerical experiments, we show that the presented solution strategy is robust w.r.t. problem and discretization parameters, efficient and scalable.
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| 5. |
- Kirby, Robert C., et al.
(författare)
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Common and Unusual Finite Elements
- 2012
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Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 95-119
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- This chapter provides a glimpse of the considerable range of finite elements in the literature. Many of the elements presented here are implemented as part of the FEniCS project already; some are future work. The universe of finite elements extends far beyond what we consider here. In particular, we consider only simplicial, polynomial-based elements. We thus bypass elements defined on quadrilaterals and hexahedra, composite and macro-element techniques, as well as XFEM-type methods. Even among polynomial-based elements on simplices, the list of elements can be extended. Nonetheless, this chapter presents a comprehensive collection of some the most common, and some more unusual, finite elements.
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| 6. |
- Logg, Anders, 1976, et al.
(författare)
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FFC: the FEniCS Form Compiler
- 2012
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Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 227-238
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- One of the key features of FEniCS is automated code generation for the general and efficient 7018 solution of finite element variational problems. This automated code generation relies on a form 7019 compiler for offline or just-in-time compilation of code for individual forms. Two different form 7020 compilers are available as part of FEniCS.
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| 7. |
- Massing, André, et al.
(författare)
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A Nitsche-Based Cut Finite Element Method for a Fluid--Structure Interaction Problem
- 2015
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Ingår i: Communications in Applied Mathematics and Computational Science. - : Mathematical Sciences Publishers. - 1559-3940 .- 2157-5452. ; 10:2, s. 97-120
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Tidskriftsartikel (refereegranskat)abstract
- We present a new composite mesh finite element method for fluid-structure interaction problems. The method is based on surrounding the structure by a boundary-fitted fluid mesh that is embedded into a fixed background fluid mesh. The embedding allows for an arbitrary overlap of the fluid meshes. The coupling between the embedded and background fluid meshes is enforced using a stabilized Nitsche formulation that allows us to establish stability and optimal-order a priori error estimates. We consider here a steady state fluid-structure interaction problem where a hyperelastic structure interacts with a viscous fluid modeled by the Stokes equations. We evaluate an iterative solution procedure based on splitting and present three-dimensional numerical examples.
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| 8. |
- Massing, Andre, et al.
(författare)
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A Stabilized Nitsche Fictitious Domain Method for the Stokes Problem
- 2014
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Ingår i: Journal of Scientific Computing. - : Springer Science and Business Media LLC. - 0885-7474 .- 1573-7691. ; 61:3, s. 604-628
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Tidskriftsartikel (refereegranskat)abstract
- We present a novel finite element method for the Stokes problem on fictitious domains. We prove inf-sup stability, optimal order convergence and uniform boundedness of the condition number of the discrete system. The finite element formulation is based on a stabilized Nitsche method with ghost penalties for the velocity and pressure to obtain stability in the presence of small cut elements. We demonstrate for the first time the applicability of the Nitsche fictitious domain method to three-dimensional Stokes problems. We further discuss a general, flexible and freely available implementation of the method and present numerical examples supporting the theoretical results.
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| 9. |
- Vinje, Vegard, et al.
(författare)
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Intracranial pressure elevation alters CSF clearance pathways
- 2020
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Ingår i: Fluids and Barriers of the CNS. - : BioMed Central. - 2045-8118. ; 17:1
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Tidskriftsartikel (refereegranskat)abstract
- Background: Infusion testing is a common procedure to determine whether shunting will be beneficial in patients with normal pressure hydrocephalus. The method has a well-developed theoretical foundation and corresponding mathematical models that describe the CSF circulation from the choroid plexus to the arachnoid granulations. Here, we investigate to what extent the proposed glymphatic or paravascular pathway (or similar pathways) modifies the results of the traditional mathematical models.Methods: We used a compartment model to estimate pressure in the subarachnoid space and the paravascular spaces. For the arachnoid granulations, the cribriform plate and the glymphatic circulation, resistances were calculated and used to estimate pressure and flow before and during an infusion test. Finally, different variations to the model were tested to evaluate the sensitivity of selected parameters.Results: At baseline intracranial pressure (ICP), we found a very small paravascular flow directed into the subarachnoid space, while 60% of the fluid left through the arachnoid granulations and 40% left through the cribriform plate. However, during the infusion, 80% of the fluid left through the arachnoid granulations, 20% through the cribriform plate and flow in the PVS was stagnant. Resistance through the glymphatic system was computed to be 2.73 mmHg/(mL/min), considerably lower than other fluid pathways, giving non-realistic ICP during infusion if combined with a lymphatic drainage route.Conclusions: The relative distribution of CSF flow to different clearance pathways depends on ICP, with the arachnoid granulations as the main contributor to outflow. As such, ICP increase is an important factor that should be addressed when determining the pathways of injected substances in the subarachnoid space. Our results suggest that the glymphatic resistance is too high to allow for pressure driven flow by arterial pulsations and at the same time too small to allow for a direct drainage route from PVS to cervical lymphatics.
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