| 31. |
- Kirby, Robert C., et al.
(författare)
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FErari: an Optimizing Compiler for Variational Forms
- 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. 239-246
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- In Chapter 8, we presented a framework for efficient evaluation of multilinear forms based on 7311 expressing the multilinear form as a special tensor contraction. This allows generation of efficient 7312 low-level code for assembly of a range of multilinear forms.
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| 32. |
- Kirby, Robert C., et al.
(författare)
-
Finite Element Variational Forms
- 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. 133-140
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- Much of the FEniCS software is devoted to the formulation of variational forms (UFL), the discretization of variational forms (FIAT, FFC, SyFi) and the assembly of the corresponding discrete operators (UFC, DOLFIN). This chapter summarizes the notation for variational forms used throughout FEniCS.
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| 33. |
- Kirby, R.C., et al.
(författare)
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Optimizing the evaluation of finite element matrices
- 2006
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 27:3, s. 741-758
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Tidskriftsartikel (refereegranskat)abstract
- Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce the cost of building local stiffness matrices for the Laplace operator and for the trilinear form for Navier-Stokes operators. For the Laplace operator in two space dimensions, we have developed a heuristic graph algorithm that searches for such redundancies and generates code for computing the local stiffness matrices. Up to cubics, we are able to build the stiffness matrix on any triangle in less than one multiply-add pair per entry. Up to sixth degree, we can do it in less than about two pairs. Preliminary low-degree results for Poisson and Navier-Stokes operators in three dimensions are also promising. © 2005 Society for Industrial and Applied Mathematics.
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| 34. |
- Kirby, Robert C., et al.
(författare)
-
Tensor Representation of Finite Element Variational Forms
- 2012
-
Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 159-162
-
Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- In Chapter 6, we saw that an important step in the assembly of matrices and vectors for the 4523 discretization of finite element variational problems is the evaluation of the cell (element) tensor AT 4524 defined by 4525 AT,i = aT(f T,r ir,...,fT,2 i2,fT,1 i1 ).
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| 35. |
- Kirby, Robert C., et al.
(författare)
-
The Finite Element Method
- 2012
-
Ingår i: Automated Solution of Differential Equations by the Finite Element Method. Anders Logg, Kent-Andre Mardal, Garth Wells (Eds.). - Heidelberg : Springer. - 9783642230981 ; , s. 77-94
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Bokkapitel (övrigt vetenskapligt/konstnärligt)abstract
- The finite element method has emerged as a universal method for the solution of differential equations. Much of the success of the finite element method can be attributed to its generality and elegance, allowing a wide range of differential equations from all areas of science to be analyzed and solved within a common framework. Another contributing factor to the success of the finite element method is the flexibility of formulation, allowing the properties of the discretization to be controlled by the choice of approximating finite element spaces.
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| 36. |
- Kirby, R.C., et al.
(författare)
-
Topological optimization of the evaluation of finite element matrices
- 2006
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Ingår i: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 28:1, s. 224-240
-
Tidskriftsartikel (refereegranskat)abstract
- We present a topological framework for finding low-flop algorithms for evaluating element stiffness matrices associated with multilinear forms for finite element methods posed over straight-sided affine domains. This framework relies on phrasing the computation on each element as the contraction of each collection of reference element tensors with an element-specific geometric tensor. We then present a new concept of complexity-reducing relations that serve as distance relations between these reference element tensors. This notion sets up a graph-theoretic context in which we may find an optimized algorithm by computing a minimum spanning tree. We present experimental results for some common multilinear forms showing significant reductions in operation count and also discuss some efficient algorithms for building the graph we use for the optimization. © 2006 Society for Industrial and Applied Mathematics.
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| 37. |
- Kolibarov, N., et al.
(författare)
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Roof Segmentation Towards Digital Twin Generation in LoD2+Using Deep Learning
- 2022
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Ingår i: IFAC-PapersOnLine. - : Elsevier BV. - 2405-8963 .- 2405-8963. ; 55:11, s. 173-178
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Konferensbidrag (refereegranskat)abstract
- There is an increasing need for digital twins of cities and their base maps, 3D city models. Creating and updating these twins is not an easy task, so automating and streamlining the process is a field of active research. A significant part of the urban geometry is residential buildings and their roofs. Modeling of roofs for urban buildings can be divided into three main areas - building detection, roof recognition and building reconstruction. The building and roofs are segmented with the help of machine learning and image processing. Afterwards the extracted information is used to generate parametric models for the roofs using methods from computational geometry. The goal is to create correct virtual models of roofs belonging to many different types of buildings. In this study, a supervised deep learning approach is proposed for the segmentation of roof edges from a single orthophoto. The predicted features include the linear elements of roofs. The experiments show that, despite the small amount of training data, even in the presence of noise, the proposed method performs well on semantic segmentation of roofs with different shapes and complexities. The quality of the extracted roof elements for the test area is about 56% and 71% for mean intersection over union (IOU) and Dice metric scores, respectively. Copyright (C) 2022 The Authors.
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| 38. |
- Langtangen, Hans Petter, et al.
(författare)
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Solving PDEs in Python
- 2016
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- This book offers a concise and gentle introduction to finite element programming in Python based on the popular FEniCS software library. Using a series of examples, including the Poisson equation, the equations of linear elasticity, the incompressible Navier–Stokes equations, and systems of nonlinear advection–diffusion–reaction equations, it guides readers through the essential steps to quickly solving a PDE in FEniCS, such as how to define a finite variational problem, how to set boundary conditions, how to solve linear and nonlinear systems, and how to visualize solutions and structure finite element Python programs.
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| 39. |
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| 40. |
- Larson, Mats G., et al.
(författare)
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Space-time CutFEM on overlapping meshes I: simple continuous mesh motion
- 2024
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Ingår i: NUMERISCHE MATHEMATIK. - : Springer Science+Business Media B.V.. - 0029-599X .- 0945-3245. ; 156, s. 1015-1054
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Tidskriftsartikel (refereegranskat)abstract
- We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/"on top" of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche's method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.
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