| 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 S., et al.
(författare)
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UFC: a Finite Element Code Generation Interface
- 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. 283-302
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- A central component of FEniCS is the UFC interface (Unified Form-assembly Code). UFC is an 8703 interface between problem-specific and general-purpose components of finite element programs. In 8704 particular, the UFC interface defines the structure and signature of the code that is generated by 8705 the form compilers FFC and SFC for DOLFIN. The UFC interface applies to a wide range of finite 8706 element problems (including mixed finite elements and discontinuous Galerkin methods) and may be 8707 used with libraries that differ widely in their design.
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| 3. |
- Alnæs, M.S., et al.
(författare)
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Unified framework for finite element assembly
- 2009
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Ingår i: International Journal of Computational Science and Engineering. - 1742-7185 .- 1742-7193. ; 4:4, s. 231-244
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Tidskriftsartikel (refereegranskat)abstract
- At the heart of any finite element simulation is the assembly of matrices and vectors from discrete variational forms. We propose a general interface between problem-specific and general-purpose components of finite element programs. This interface is called Unified Form-assembly Code (UFC). A wide range of finite element problems is covered, including mixed finite elements and discontinuous Galerkin methods. We discuss how the UFC interface enables implementations of variational form evaluation to be independent of mesh and linear algebra components. UFC does not depend on any external libraries, and is released into the public domain. Copyright © 2009, Inderscience Publishers.
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| 4. |
- 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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| 5. |
- Ames, Ellery, et al.
(författare)
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Cosmic string and black hole limits of toroidal Vlasov bodies in general relativity
- 2019
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Ingår i: Physical Review D. - : AMER PHYSICAL SOC. - 2470-0010 .- 2470-0029. ; 99:2
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Tidskriftsartikel (refereegranskat)abstract
- We numerically investigate limits of a two-parameter family of stationary solutions to the Einstein-Vlasov system. The solutions are toroidal and have nonvanishing angular momentum. As the parameters are tuned to more relativistic solutions (measured e.g., by an increasing redshift) we provide evidence for a sequence of solutions which approaches the extreme Kerr black hole family. Solutions with angular momentum larger than the square of the mass are also investigated, and in the relativistic limit the near-field geometry of such solutions is observed to become locally rotationally symmetric about the matter density. The existence of a deficit angle in these regions is investigated.
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| 6. |
- Ames, Ellery, 1984, et al.
(författare)
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On axisymmetric and stationary solutions of the self-gravitating Vlasov system
- 2016
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Ingår i: Classical and Quantum Gravity. - : IOP Publishing. - 0264-9381 .- 1361-6382. ; 33:15
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Tidskriftsartikel (refereegranskat)abstract
- Axisymmetric and stationary solutions are constructed to the Einstein-Vlasov and Vlasov-Poisson systems. These solutions are constructed numerically, using finite element methods and a fixed-point iteration in which the total mass is fixed at each step. A variety of axisymmetric stationary solutions are exhibited, including solutions with toroidal, disk-like, spindle-like, and composite spatial density configurations, as are solutions with non-vanishing net angular momentum. In the case of toroidal solutions, we show for the first time, solutions of the Einstein-Vlasov system which contain ergoregions.
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| 7. |
- Arnold, Douglas N., et al.
(författare)
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Periodic Table of the Finite Elements
- 2014
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Ingår i: SIAM News. - 0036-1437. ; 47:9
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Tidskriftsartikel (refereegranskat)abstract
- The finite element method is one of the most powerful and widely applicable techniques for the numerical solution of partial differential equations and, therefore, for the simulation of the physical world. First proposed by engineers in the 1950s as a practical numerical method for predicting the deflection and stress of structural components of aircraft, the method has since been continuously extended and refined. It is now used in almost all application areas modeled by PDEs: solid and fluid dynamics, electromagnetics, biophysics, and even finance, to name just a few. Much as the chemical elements can be arranged in a periodic table based on their electron structure and recurring chemical properties, a broad assortment of finite elements can be arranged in a table that clarifies their properties and relationships. This arrangement, which is based on expression of the finite element function spaces in the language of differential forms, is one of the major outcomes of the theory known as finite element exterior calculus, or FEEC. Just as the arrangement of the chemical elements in a periodic table led to the discovery of new elements, the periodic table of finite elements has not only clarified existing elements but also highlighted holes in our knowledge and led to new families of finite elements suited for certain purposes.
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| 8. |
- 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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| 9. |
- Borgqvist, Johannes, 1990, et al.
(författare)
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Cell polarisation in a bulk-surface model can be driven by both classic and non-classic Turing instability
- 2021
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Ingår i: Npj Systems Biology and Applications. - : Springer Science and Business Media LLC. - 2056-7189. ; 7:1
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Tidskriftsartikel (refereegranskat)abstract
- The GTPase Cdc42 is the master regulator of eukaryotic cell polarisation. During this process, the active form of Cdc42 is accumulated at a particular site on the cell membrane called the pole. It is believed that the accumulation of the active Cdc42 resulting in a pole is driven by a combination of activation-inactivation reactions and diffusion. It has been proposed using mathematical modelling that this is the result of diffusion-driven instability, originally proposed by Alan Turing. In this study, we developed, analysed and validated a 3D bulk-surface model of the dynamics of Cdc42. We show that the model can undergo both classic and non-classic Turing instability by deriving necessary conditions for which this occurs and conclude that the non-classic case can be viewed as a limit case of the classic case of diffusion-driven instability. Using three-dimensional Spatio-temporal simulation we predicted pole size and time to polarisation, suggesting that cell polarisation is mainly driven by the reaction strength parameter and that the size of the pole is determined by the relative diffusion.
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| 10. |
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| 11. |
- Eriksson, Kenneth, 1952, et al.
(författare)
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Explicit time-stepping for stiff ODES
- 2003
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Ingår i: SIAM Journal on Scientific Computing. - 1064-8275 .- 1095-7197. ; 25:4, s. 1142-1157
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Tidskriftsartikel (refereegranskat)abstract
- We present a new strategy for solving stiff ODEs with explicit methods. By adaptively taking a small number of stabilizing small explicit time steps when necessary, a stiff ODE system can be stabilized enough to allow for time steps much larger than what is indicated by classical stability analysis. For many stiff problems the cost of the stabilizing small time steps is small, so the improvement is large. We illustrate the technique on a number of well-known stiff test problems.
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| 12. |
- Gonzalez Caceres, Alex Arnoldo, 1982, et al.
(författare)
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Towards digital twinning for multi-domain simulation workflows in urban design: a case study in Gothenburg
- 2024
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Ingår i: Journal of Building Performance Simulation. - 1940-1493 .- 1940-1507. ; In Press
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Tidskriftsartikel (refereegranskat)abstract
- This work proposes an automated workflow using digital twinning for multi-domain environmental performance analysis of urban developments. Digital twins can potentially provide a common basis for multi-domain simulations and help overcome data availability and interoperability issues. The proposed workflow consists of five steps: (1) creating a procedural urban 3D model, (2) generating design alternatives parametrically, (3) exporting the context and each design alternative to each simulation tool, (4) running simulations for wind comfort, energy demand, and noise for each design alternative, and (5) combining and visualizing the simulation results using the digital twin. The workflow was applied to a neighbourhood in Sweden, the resultsreveal significant reduction in manual work when applying multiple simulation software for different domains. This is one step forward in streamlining the workflow for urban analysis, crucial for multi-domain optimization. In the future, further domains and simulation tools can be added to the workflow.
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| 13. |
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| 14. |
- Hoffman, Johan, 1974, et al.
(författare)
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Mathematics and Computation
- 2004
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Ingår i: Stockholm Intelligencer: Fourth European Congress of Mathematics.
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Konferensbidrag (övrigt vetenskapligt/konstnärligt)
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| 15. |
- Jansson, Johan, et al.
(författare)
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Algorithms and Data Structures for Multi-Adaptive Time-Stepping
- 2008
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Ingår i: ACM Transactions on Mathematical Software. - : Association for Computing Machinery (ACM). - 0098-3500 .- 1557-7295. ; 35:3
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Tidskriftsartikel (refereegranskat)abstract
- Multi-adaptive Galerkin methods are extensions of the standard continuous and discontinuous Galerkin methods for the numerical solution of initial value problems for ordinary or partial differential equations. In particular, the multi-adaptive methods allow individual and adaptive time steps to be used for different components or in different regions of space. We present algorithms for efficient multi-adaptive time-stepping, including the recursive construction of time slabs and adaptive time step selection. We also present data structures for efficient storage and interpolation of the multi-adaptive solution. The efficiency of the proposed algorithms and data structures is demonstrated for a series of benchmark problems.
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| 16. |
- Jansson, Johan, 1978, et al.
(författare)
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Computational modeling of dynamical systems
- 2005
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Ingår i: Mathematical Models and Methods in Applied Sciences. - 0218-2025. ; 15:3, s. 471-481
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Tidskriftsartikel (refereegranskat)abstract
- In this short note, we discuss the basic approach to computational modeling of dynamical systems. If a dynamical system contains multiple time scales, ranging from very fast to slow, computational solution of the dynamical system can be very costly. By resolving the fast time scales in a short time simulation, a model for the effect of the small time scale variation on large time scales can be determined, making solution possible on a long time interval. This process of computational modeling can be completely automated. Two examples are presented, including a simple model problem oscillating at a time scale of 10-9 computed over the time interval [0, 100], and a lattice consisting of large and small point masses. © World Scientific Publishing Company.
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| 17. |
- Johansson, August, et al.
(författare)
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A Multimesh Finite Element Method for the Stokes Problem
- 2020
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Ingår i: Lecture Notes in Computational Science and Engineering. - Cham : Springer International Publishing. - 1439-7358 .- 2197-7100. ; 132, s. 189-198, s. 189-198
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Konferensbidrag (refereegranskat)abstract
- The multimesh finite element method enables the solution of partial differential equations on a computational mesh composed by multiple arbitrarily overlapping meshes. The discretization is based on a continuous–discontinuous function space with interface conditions enforced by means of Nitsche’s method. In this contribution, we consider the Stokes problem as a first step towards flow applications. The multimesh formulation leads to so called cut elements in the underlying meshes close to overlaps. These demand stabilization to ensure coercivity and stability of the stiffness matrix. We employ a consistent least-squares term on the overlap to ensure that the inf-sup condition holds. We here present the method for the Stokes problem, discuss the implementation, and verify that we have optimal convergence.
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| 18. |
- Johansson, August, et al.
(författare)
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High order cut finite element methods for the Stokes problem
- 2015
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Ingår i: Advanced Modeling and Simulation in Engineering Sciences. - : Springer. - 2213-7467. ; 2:1, s. 1-23
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Tidskriftsartikel (refereegranskat)abstract
- We develop a high order cut finite element method for the Stokes problem based on general inf-sup stable finite element spaces. We focus in particular on composite meshes consisting of one mesh that overlaps another. The method is based on a Nitsche formulation of the interface condition together with a stabilization term. Starting from inf-sup stable spaces on the two meshes, we prove that the resulting composite method is indeed inf-sup stable and as a consequence optimal a priori error estimates hold.
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| 19. |
- Johansson, A., et al.
(författare)
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Multimesh finite element methods: Solving PDEs on multiple intersecting meshes
- 2019
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 343, s. 672-689
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Tidskriftsartikel (refereegranskat)abstract
- We present a new framework for expressing finite element methods on multiple intersecting meshes: multimesh finite element methods. The framework enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Such multimesh finite element methods are particularly well suited to problems in which the computational domain undergoes large deformations as a result of the relative motion of the separate components of a multi-body system. In the present paper, we formulate the multimesh finite element method for the Poisson equation. Numerical examples demonstrate the optimal order convergence, the numerical robustness of the formulation and implementation in the face of thin intersections and rounding errors, as well as the applicability of the methodology. In the accompanying paper (Johansson et al., 2018), we analyze the proposed method and prove optimal order convergence and stability. (C) 2018 Elsevier B.V. All rights reserved.
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| 20. |
- Johansson, August, et al.
(författare)
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Multimesh finite elements with flexible mesh sizes
- 2020
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 372
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Tidskriftsartikel (refereegranskat)abstract
- We analyze a new framework for expressing finite element methods on arbitrarily many intersecting meshes: multimesh finite element methods. The multimesh finite element method, first presented in Johansson et al. (2019), enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Furthermore, each of these meshes may have its own mesh parameter. In the present paper we study the Poisson equation and show that the proposed formulation is stable without assumptions on the relative sizes of the mesh parameters. In particular, we prove optimal order a priori error estimates as well as optimal order estimates of the condition number. Throughout the analysis, we trace the dependence of the number of intersecting meshes. Numerical examples are included to illustrate the stability of the method.
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| 21. |
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| 22. |
- Kehlet, Benjamin, et al.
(författare)
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A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations
- 2017
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Ingår i: Numerical Algorithms. - : Springer Science and Business Media LLC. - 1017-1398 .- 1572-9265. ; 76:1, s. 191-210
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Tidskriftsartikel (refereegranskat)abstract
- © 2016, The Author(s). We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretization errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator.
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| 23. |
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| 24. |
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| 25. |
- Ketzler, Bernd, 1982, et al.
(författare)
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Digital Twins for Cities: A State of the Art Review
- 2020
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Ingår i: Built Environment. - : Alexandrine Press. - 0263-7960. ; 46:4, s. 547-573
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Tidskriftsartikel (refereegranskat)abstract
- During the last decades, a variety of digital tools have been developed to support both the planning and management of cities, as well as the inclusion of civic society. Here, the concept of a Digital Twin – which is rapidly emerging throughout many disciplines due to advances in technology, computational capacities and availability of large amounts of data – plays an important role. In short, a digital twin is a living virtual model, a connected digital representation of a physical system and has been a central concept in the manufacturing industry for the past decades. In this article, we review the terminology of digital twins for cities and identify commonalities and relations to the more established term 3D city models. Our fi ndings indicate an increasing use of the term digital twin in academic literature, both in general and in the context of cities and the built environment. We fi nd that while there is as yet no consensus on the exact defi nition of what constitutes a digital twin, it is increasingly being used to describe something that is more than a 3D city model (including, e.g. semantic data, real-time sensor data, physical models, and simulations). At the same time, the term has not yet replaced the term 3D city model as the most dominant term in the 3D GIS domain. By looking at grey literature we discuss how digital twins for cities are implemented in practice and present examples of digital twins in a global perspective. Further, we discuss some of the application areas and potential challenges for future development and implementation of digital twins for cities. We conclude that there are signifi cant opportunities for up-scaling digital twins, with the potential to bring benefi ts to the city and its citizens and clients.
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