| 6021. |
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| 6022. |
- Hansbo, Peter, et al.
(författare)
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Analysis of finite element methods for vector Laplacians on surfaces
- 2020
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Ingår i: IMA Journal of Numerical Analysis. - : Oxford University Press. - 0272-4979 .- 1464-3642. ; 40:3, s. 1652-1701
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Tidskriftsartikel (refereegranskat)abstract
- We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in R-3. Closely related operators arise in models of flow on surfaces as well as elastic membranes and shells. The method is based on standard continuous parametric Lagrange elements that describe a R-3 vector field on the surface, and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take into account the approximation of both the geometry of the surface and the solution to the partial differential equation. In particular, we note that to achieve optimal order error estimates, in both energy and L-2 norms, the normal approximation used in the penalization term must be of the same order as the approximation of the solution. This can be fulfilled either by using an improved normal in the penalization term, or by increasing the order of the geometry approximation. We also present numerical results using higher-order finite elements that verify our theoretical findings.
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| 6023. |
- Hansbo, Peter, et al.
(författare)
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Augmented Lagrangian approach to deriving discontinuous Galerkin methods for nonlinear elasticity problems
- 2022
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Ingår i: International Journal for Numerical Methods in Engineering. - : John Wiley & Sons. - 0029-5981 .- 1097-0207. ; 123:18, s. 4407-4421
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Tidskriftsartikel (refereegranskat)abstract
- We use the augmented Lagrangian formalism to derive discontinuous Galerkin (DG) formulations for problems in nonlinear elasticity. In elasticity, stress is typically a symmetric function of strain, leading to symmetric tangent stiffness matrices in Newton's method when conforming finite elements are used for discretization. By use of the augmented Lagrangian framework, we can also obtain symmetric tangent stiffness matrices in DG methods. We suggest two different approaches and give examples from plasticity and from large deformation hyperelasticity.
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| 6024. |
- Hansbo, Peter, et al.
(författare)
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Characteristic cut finite element methods for convection-diffusion problems on time dependent surfaces
- 2015
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 293, s. 431-461
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Tidskriftsartikel (refereegranskat)abstract
- We develop a finite element method for convection-diffusion problems on a given time dependent surface, for instance modeling the evolution of a surfactant. The method is based on a characteristic-Galerkin formulation combined with a piecewise linear cut finite element method in space. The cut finite element method is constructed by embedding the surface in a background grid and then using the restriction to the surface of a finite element space defined on the background grid. The surface is allowed to cut through the background grid in an arbitrary fashion. To ensure stability and well posedness of the resulting algebraic systems of equations, independent of the position of the surface in the background grid, we add a consistent stabilization term. We prove error estimates and present confirming numerical results. (C) 2015 Elsevier B.V. All rights reserved.
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| 6025. |
- Hansbo, Peter, 1959-, et al.
(författare)
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Continuous/discontinuous finite element modelling of Kirchhoff plate structures in R3 using tangential differential calculus
- 2017
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Ingår i: Computational Mechanics. - : Springer. - 0178-7675 .- 1432-0924. ; 60:4, s. 693-702
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Tidskriftsartikel (refereegranskat)abstract
- We employ surface differential calculus to derive models for Kirchhoff plates including in-plane membrane deformations. We also extend our formulation to structures of plates. For solving the resulting set of partial differential equations, we employ a finite element method based on elements that are continuous for the displacements and discontinuous for the rotations, using (Formula presented.)-elements for the discretisation of the plate as well as for the membrane deformations. Key to the formulation of the method is a convenient definition of jumps and averages of forms that are d-linear in terms of the element edge normals.
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| 6026. |
- Hansbo, Peter, 1959-, et al.
(författare)
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Cut Finite Element Methods for Linear Elasticity Problems
- 2017
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Ingår i: Geometrically Unfitted Finite Element Methods and Applications. - Cham : Springer International Publishing. - 9783319714301 - 9783319714318 ; , s. 25-63
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Konferensbidrag (refereegranskat)abstract
- We formulate a cut finite element method for linear elasticity based on higher order elements on a fixed background mesh. Key to the method is a stabilization term which provides control of the jumps in the derivatives of the finite element functions across faces in the vicinity of the boundary. We then develop the basic theoretical results including error estimates and estimates of the condition number of the mass and stiffness matrices. We apply the method to the standard displacement problem, the frequency response problem, and the eigenvalue problem. We present several numerical examples including studies of thin bending dominated structures relevant for engineering applications. Finally, we develop a cut finite element method for fibre reinforced materials where the fibres are modeled as a superposition of a truss and a Euler-Bernoulli beam. The beam model leads to a fourth order problem which we discretize using the restriction of the bulk finite element space to the fibre together with a continuous/discontinuous finite element formulation. Here the bulk material stabilizes the problem and it is not necessary to add additional stabilization terms.
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| 6027. |
- Hansbo, Peter, et al.
(författare)
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Discontinuous Galerkin and the Crouzeix-Raviart element : Application to elasticity
- 2003
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Ingår i: Mathematical Modelling and Numerical Analysis. - : EDP Sciences. - 0764-583X .- 1290-3841. ; 37:1, s. 63-72
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Tidskriftsartikel (refereegranskat)abstract
- We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix-Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and ( nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.
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| 6028. |
- Hansbo, Peter, et al.
(författare)
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Discontinuous Galerkin methods for incompressible and nearly incompressible elasticity by Nitsche's method
- 2002
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - 0045-7825 .- 1879-2138. ; 191:17-18, s. 1895-1908
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Tidskriftsartikel (refereegranskat)abstract
- We propose and analyze a discontinuous finite element method for nearly incompressible linear elasticity on triangular meshes. We show optimal error estimates that are uniform with respect to Poisson's ratio. The method is thus locking free. We also introduce an equivalent mixed formulation, allowing for completely incompressible elasticity problems. Numerical results are presented.
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| 6029. |
- Hansbo, Peter F G, 1959
(författare)
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A discontinuous finite element method for elasto-plasticity
- 2010
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Ingår i: International Journal for Numerical Methods in Biomedical Engineering. - : Wiley. - 2040-7939 .- 2040-7947. ; 26:6, s. 780-789
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Tidskriftsartikel (refereegranskat)abstract
- We propose an interior penalty discontinuous finite element method for small strain elasto-plasticity using triangular or tetrahedral meshes. A new penalty formulation suitable for plasticity, in particular allowing for inter-element slip, is introduced. The method is also locking free, which is crucial as the plastic zone may exhibit an incompressible response. Numerical results are presented.
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| 6030. |
- Hansbo, Peter F G, 1959, et al.
(författare)
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A discontinuous Galerkin method for the plate equation
- 2002
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Ingår i: Calcolo. - : Springer Science and Business Media LLC. - 0008-0624 .- 1126-5434. ; 39:1, s. 41-59
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Tidskriftsartikel (refereegranskat)abstract
- We present a discontinuous Galerkin method for the plate problem. The method employs a discontinuous approximation space allowing nonmatching grids and different types of approximation spaces. Continuity is enforced weakly through the variational form. Discrete approximations of the normal and twisting moments and the transversal force, which satisfy the equilibrium condition on an element level, occur naturally in the method. We show optimal a priori error estimates in various norms and investigate locking phenomena when certain stabilization parameters tend to infinity. Finally, we relate the method to two classical elements: the nonconforming Morley element and the C^1 Argyris element.
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