scholarly journals Trace finite element methods for surface vector-Laplace equations

2020 ◽  
Author(s):  
Thomas Jankuhn ◽  
Arnold Reusken
Keyword(s):  
Finite Element ◽  
Lagrange Multiplier ◽  
Finite Element Methods ◽  
Penalty Method ◽  
Penalty Parameter ◽  
Element Discretization ◽  
Level Set Function ◽  
Laplace Equations ◽  
Set Function ◽  
Unknown Vector

Abstract In this paper we analyze a class of trace finite element methods for the discretization of vector-Laplace equations. A key issue in the finite element discretization of such problems is the treatment of the constraint that the unknown vector field must be tangential to the surface (‘tangent condition’). We study three different natural techniques for treating the tangent condition, namely a consistent penalty method, a simpler inconsistent penalty method and a Lagrange multiplier method. The main goal of the paper is to present an analysis that reveals important properties of these three different techniques for treating the tangent constraint. A detailed error analysis is presented that takes the approximation of both the geometry of the surface and the solution of the partial differential equation into account. Error bounds in the energy norm are derived that show how the discretization error depends on relevant parameters such as the degree of the polynomials used for the approximation of the solution, the degree of the polynomials used for the approximation of the level set function that characterizes the surface, the penalty parameter and the degree of the polynomials used for the approximation of the Lagrange multiplier.

scholarly journals Consistent coupling of positions and rotations for embedding 1D Cosserat beams into 3D solid volumes

2021 ◽  
Author(s):  
Ivo Steinbrecher ◽  
Alexander Popp ◽  
Christoph Meier
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Lagrange Multiplier ◽  
Degrees Of Freedom ◽  
Element Discretization ◽  
Geometrically Exact Beam ◽  
Coupling Constraints ◽  
Rotational Degrees Of Freedom ◽  
3D Solid ◽  
Cosserat Continua

AbstractThe present article proposes a mortar-type finite element formulation for consistently embedding curved, slender beams into 3D solid volumes. Following the fundamental kinematic assumption of undeformable cross-section s, the beams are identified as 1D Cosserat continua with pointwise six (translational and rotational) degrees of freedom describing the cross-section (centroid) position and orientation. A consistent 1D-3D coupling scheme for this problem type is proposed, requiring to enforce both positional and rotational constraints. Since Boltzmann continua exhibit no inherent rotational degrees of freedom, suitable definitions of orthonormal triads are investigated that are representative for the orientation of material directions within the 3D solid. While the rotation tensor defined by the polar decomposition of the deformation gradient appears as a natural choice and will even be demonstrated to represent these material directions in a $$L_2$$ L 2 -optimal manner, several alternative triad definitions are investigated. Such alternatives potentially allow for a more efficient numerical evaluation. Moreover, objective (i.e. frame-invariant) rotational coupling constraints between beam and solid orientations are formulated and enforced in a variationally consistent manner based on either a penalty potential or a Lagrange multiplier potential. Eventually, finite element discretization of the solid domain, the embedded beams, which are modeled on basis of the geometrically exact beam theory, and the Lagrange multiplier field associated with the coupling constraints results in an embedded mortar-type formulation for rotational and translational constraint enforcement denoted as full beam-to-solid volume coupling (BTS-FULL) scheme. Based on elementary numerical test cases, it is demonstrated that a consistent spatial convergence behavior can be achieved and potential locking effects can be avoided, if the proposed BTS-FULL scheme is combined with a suitable solid triad definition. Eventually, real-life engineering applications are considered to illustrate the importance of consistently coupling both translational and rotational degrees of freedom as well as the upscaling potential of the proposed formulation. This allows the investigation of complex mechanical systems such as fiber-reinforced composite materials, containing a large number of curved, slender fibers with arbitrary orientation embedded in a matrix material.

Duality-based adaptivity in finite element discretization of heterogeneous multiscale problems

2016 ◽  
Vol 24 (3) ◽  
Author(s):  
Matthias Maier ◽  
Rolf Rannacher
Keyword(s):  
Finite Element ◽  
Finite Element Methods ◽  
Elliptic Problems ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Multiscale Problems ◽  
Multiscale Finite Element ◽  
Heterogeneous Multiscale ◽  
Multiscale Finite Element Methods

AbstractThis paper introduces an framework for adaptivity for a class of heterogeneous multiscale finite element methods for elliptic problems, which is suitable for

scholarly journals Revised Level Set-Based Method for Topology Optimization and Its Applications in Bridge Construction

2017 ◽  
Vol 11 (1) ◽  
pp. 153-166
Author(s):  
Jing Wu ◽  
Li Wu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Topology Optimization ◽  
Level Set ◽  
The Finite Element Method ◽  
Bridge Design ◽  
Bridge Construction ◽  
Level Set Function ◽  
Set Function ◽  
Element Method

To cure imperfections such as low accuracy and the lack of ability to nucleate hole in the conventional level set-based topology optimization method, a novel method using a trapezoidal method with discrete design variables is proposed. The proposed method can simultaneously accomplish topology and shape optimization. The finite element method is employed to obtain element properties and provide data for calculating design and topological sensitivities. With the aim of performing the finite element method on a non-conforming mesh, a relation between the level set function and the element densities field has to be clearly defined. The element densities field is obtained by averaging the Heaviside function values. The Lagrange multiplier method is exploited to fulfill the volume constraint. Based on topological and design sensitivity and the trapezoidal method, the Hamilton-Jacobi partial differential equation is updated recursively to find the optimal layout. In order to stabilize the iterations and improve the efficiency of the algorithm, re-initiation of the level set function is necessary. Then, the detailed process of a cantilever design is illustrated. To demonstrate the applications of the proposed method in bridge construction, two numerical examples of a pylon bridge design are introduced. It is shown that the results match practical designs very well, and the proposed method is a helpful tool in bridge design.

scholarly journals The Pulley Element

2016 ◽  
Vol 16 (2) ◽  
pp. 161-164
Author(s):  
Hynek Štekbauer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Lagrange Multiplier ◽  
Penalty Method ◽  
Lagrange Multiplier Method ◽  
Multiplier Method ◽  
The Finite Element Method ◽  
Cable System ◽  
Mechanical Advantage ◽  
Element Method

Abstract The pulley is used in a number of structures for the mechanical advantage it gives. This paper presents an approach for the calculation of a pulley-cable system using a special pulley element in the finite element method. The Lagrange Multiplier method and Penalty method are used to define the pulley element, as described in this paper. Both approaches are easy to implement in general FEM codes.

On the Use of the Level Set Method With Unstructured Grids

2001 ◽  
Author(s):  
Sergey V. Shepel ◽  
Samuel Paolucci
Keyword(s):  
Finite Element ◽  
Level Set Method ◽  
Level Set ◽  
Unstructured Grids ◽  
Finite Element Formulation ◽  
Element Formulation ◽  
Level Set Function ◽  
Complicated Geometry ◽  
Rkdg Finite Element Method ◽  
Set Function

Abstract A mass preserving finite element formulation of the Level Set method is presented. The formulation is based on the discontinuous representation of the level set function and involves the Runge-Kutta Discontinuous Galerkin (RKDG) finite element method. The resulting formulation has the flexibility of treating a complicated geometry by using arbitrary triangulation. The performance of the scheme is demonstrated on a number of two-dimensional re-distance and coupled advection-redistance problems. The results indicate that the RKDG finite element formulation provides accurate solutions of the Level Set problem and has great potential in fluid dynamics applications.

scholarly journals Mixed Finite Element Methods for the Poisson Equation Using Biorthogonal and Quasi-Biorthogonal Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Bishnu P. Lamichhane
Keyword(s):  
Finite Element ◽  
Poisson Equation ◽  
Lagrange Multiplier ◽  
Finite Element Methods ◽  
Mixed Finite Element ◽  
Biorthogonal Systems ◽  
A Priori Error Estimate ◽  
The Poisson Equation ◽  
Mixed Formulations ◽  
Multiplier Space

We introduce two three-field mixed formulations for the Poisson equation and propose finite element methods for their approximation. Both mixed formulations are obtained by introducing a weak equation for the gradient of the solution by means of a Lagrange multiplier space. Two efficient numerical schemes are proposed based on using a pair of bases for the gradient of the solution and the Lagrange multiplier space forming biorthogonal and quasi-biorthogonal systems, respectively. We also establish an optimal a priori error estimate for both finite element approximations.

Numerical Analysis of a Finite Element Approximation to a Level Set Model for Free-Surface Flows

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tomás Chacón Rebollo ◽  
Macarena Gómez Mármol ◽  
Isabel Sánchez Muñoz
Keyword(s):  
Finite Element ◽  
Free Surface ◽  
Level Set Method ◽  
Level Set ◽  
Free Surface Flows ◽  
Surface Flow ◽  
Optimal Error Estimates ◽  
Element Approximation ◽  
Element Discretization ◽  
Level Set Function

Abstract In this paper, we study a finite element discretization of a Level Set Method formulation of free-surface flow. We consider an Euler semi-implicit discretization in time and a Galerkin discretization of the level set function. We regularize the density and viscosity of the flow across the interface, following the Level Set Method. We prove stability in natural norms when the viscosity and density vary from one to the other layer and optimal error estimates for smooth solutions when the layers have the same density. We present some numerical tests for academic flows.

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