arXiv subjects

Anders Logg

Publications and source records attributed to Anders Logg.

At least 19 recordsLinked to original sources

arXiv2023

On procedural urban digital twin generation and visualization of large scale data

The desired outcome for urban digital twins is an automatically generated detailed 3D model of a building from aerial imagery, footprints, LiDAR, or a fusion of these. Such 3D models have applications in architecture, civil engineering, urban planning, construction, real estate, GIS, and many others. Further, the visualization of large-scale data in conjunction with the generated 3D models is often a recurring and resource-intensive task. However, a completely automated end-to-end workflow is complex, requiring many steps to achieve a high-quality visualization. Methods for building reconstruction approaches have come a long way from previously manual approaches to semi-automatic or automatic approaches. The next step after reconstructing buildings is visualizing the buildings and their context. Advances in real-time rendering using game engines have enabled the extension of building reconstruction methods to procedurally generated context generation. This paper aims to complement existing methods of 3D building generation. First, we present a literature review covering different options for procedurally generated context generation and visualization methods in-depth, focusing on workflows and data pipelines. Next, we present a semi-automated workflow that extends the building reconstruction pipeline to include procedural context generation (terrain and vegetation) using Unreal Engine and, finally, the integration of various types of large-scale urban analysis data for visualization. We conclude with a series of challenges faced in achieving such pipelines and the limitations of the current approach. The steps for a complete, end-to-end solution involve developing robust systems for building detection, rooftop recognition, and geometry generation and importing and visualizing data in the same 3D environment.

cs.GR↗

arXiv2023

Space-Time CutFEM on Overlapping Meshes: Simple Continuous Mesh Motion

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 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 new energy analysis framework 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.

math.NA↗

arXiv2023

Multi-mesh multi-objective optimization with application to a model problem in urban design

We present an application of multi-mesh finite element methods as part of a methodology for optimizing settlement layouts. By formulating a multi-objective optimization problem, we demonstrate how a given number of buildings may be optimally placed on a given piece of land with respect to both wind conditions and the view experienced from the buildings. The wind flow is modeled by a multi-mesh (cut finite element) method. This allows each building to be embedded in a boundary-fitted mesh which can be moved freely on top of a fixed background mesh. This approach enables a multitude of settlement layouts to be evaluated without the need for costly mesh generation when changing the configuration of buildings. The view is modeled by a measure that takes into account the totality of unobstructed view from the collection of buildings, and is efficiently computed by rasterization.

cs.CE↗

arXiv2022

Digital twins for city simulation: Automatic, efficient, and robust mesh generation for large-scale city modeling and simulation

The concept of creating digital twins, connected digital models of physical systems, is gaining increasing attention for modeling and simulation of whole cities. The basis for building a digital twin of a city is the generation of a 3D city model, often represented as a mesh. Creating and updating such models is a tedious process that requires manual work and considerable effort, especially in the modeling of building geometries. In the current paper, we present a novel algorithm and implementation for automatic, efficient, and robust mesh generation for large-scale city modeling and simulation. The algorithm relies on standard, publicly available data, in particular 2D cadastral maps (building footprints) and 3D point clouds obtained from aerial scanning. The algorithm generates LoD1.2 city models in the form of both triangular surface meshes, suitable for visualisation, and high-quality tetrahedral volume meshes, suitable for simulation. Our tests demonstrate good performance and scaling and indicate good avenues for further optimization based on parallelisation. The long-term goal is a generic digital twin of cities volume mesh generator that provides (nearly) real-time mesh manipulation in LoD2.x.

cs.GR↗

arXiv2020

MultiMesh Finite Elements with Flexible Mesh Sizes

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 [40], 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.

math.NA↗

arXiv2018

A MultiMesh Finite Element Method for the Stokes Problem

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.

math.NA↗

arXiv2018

On Axisymmetric and Stationary Solutions of the Self-Gravitating Vlasov System

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.

gr-qc↗

arXiv2018

Cosmic String and Black Hole Limits of Toroidal Vlasov Bodies in General Relativity

We numerically investigate limits of a two-parameter family of stationary solutions to the Einstein-Vlasov system. The solutions are toroidal and have non-vanishing angular momentum. As one tunes to more relativistic solutions (measured for example by an increasing redshift) there exists 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 conical in the sense that there is a deficit angle. Such solutions may provide self-consistent models for rotating circular cosmic strings.

gr-qc↗

arXiv2018

MultiMesh Finite Element Methods: Solving PDEs on Multiple Intersecting Meshes

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~\cite{mmfem-2}, we analyze the proposed method and prove optimal order convergence and stability.

math.NA↗

arXiv2017

Solving Poisson's Equation on the Microsoft HoloLens

We present a mixed reality application (HoloFEM) for the Microsoft HoloLens. The application lets a user define and solve a physical problem governed by Poisson's equation with the surrounding real world geometry as input data. Holograms are used to visualise both the problem and the solution. The finite element method is used to solve Poisson's equation. Solving and visualising partial differential equations in mixed reality could have potential usage in areas such as building planning and safety engineering.

cs.GR↗

arXiv2015

High Order Cut Finite Element Methods for the Stokes Problem

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 \emph{a~priori} error estimates hold.

math.NA↗

arXiv2015

A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations

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 discretisation 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.

math.NA↗

arXiv2013

A Nitsche-based cut finite element method for a fluid--structure interaction problem

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 which 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 which allows us to establish stability and optimal order \emph{a priori} error estimates, see~\cite{MassingLarsonLoggEtAl2013}. 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.

math.NA↗

arXiv2013

Quantifying the computability of the Lorenz system

It is well known that the computation of accurate trajectories of the Lorenz system is a difficult problem. Computed solutions are very sensitive to the discretization error determined by the time step size and polynomial order of the method, as well as round-off errors. In this work, we show how round-off errors limit the computability of the Lorenz system and quantify exactly the length of intervals over which solutions can be computed, expressed in terms of the floating point precision. Using adjoint-based \emph{a posteriori} error analysis techniques, we estimate the stability of computations with respect to initial data, discretization, and round-off errors, respectively. The analysis is verified by computing an accurate solution on the time interval $[0, 1000]$ using a very high order (order 200) finite element method and very high floating point precision 400 digits).

math.NA↗

arXiv2012

Unified Form Language: A domain-specific language for weak formulations of partial differential equations

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 multi-field 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.

cs.MS↗

arXiv2012

Efficient implementation of finite element methods on non-matching and overlapping meshes in 3D

In recent years, a number of finite element methods have been formulated for the solution of partial differential equations on complex geometries based on non-matching or overlapping meshes. Examples of such methods include the fictitious domain method, the extended finite element method, and Nitsche's method. In all of these methods, integrals must be computed over cut cells or subsimplices which is challenging to implement, especially in three space dimensions. In this note, we address the main challenges of such an implementation and demonstrate good performance of a fully general code for automatic detection of mesh intersections and integration over cut cells and subsimplices. As a canonical example of an overlapping mesh method, we consider Nitsche's method which we apply to Poisson's equation and a linear elastic problem.

math.NA↗

arXiv2012

A stabilized Nitsche fictitious domain method for the Stokes problem

We develop a Nitsche fictitious domain method for the Stokes problem starting from a stabilized Galerkin finite element method with low order elements for both the velocity and the pressure. By introducing additional penalty terms for the jumps in the normal velocity and pressure gradients in the vicinity of the boundary, we show that the method is inf-sup stable. As a consequence, optimal order a priori error estimates are established. Moreover, the condition number of the resulting stiffness matrix is shown to be bounded independently of the location of the boundary. We discuss a general, flexible and freely available implementation of the method in three spatial dimensions and present numerical examples supporting the theoretical results.

math.NA↗

arXiv2012

A stabilized Nitsche overlapping mesh method for the Stokes problem

We develop a Nitsche-based formulation for a general class of stabilized finite element methods for the Stokes problem posed on a pair of overlapping, non-matching meshes. By ex- tending the least-squares stabilization to the overlap region, we prove that the method is stable, consistent, and optimally convergent. To avoid an ill-conditioned linear algebra system, the scheme is augmented by a least-squares term measuring the discontinuity of the solution in the overlap region of the two meshes. As a consequence, we may prove an estimate for the condition number of the resulting stiffness matrix that is independent of the location of the interface. Finally, we present numerical examples in three spatial dimensions illustrating and confirming the theoretical results.

math.NA↗