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Peter Hansbo

Publications and source records attributed to Peter Hansbo.

At least 37 records · Page 2Linked to original sources
arXiv2020

Low Regularity Estimates for CutFEM Approximations of an Elliptic Problem with Mixed Boundary Conditions

We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution $u \in H^s$ with $s\in (1,3/2]$. For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.

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arXiv2020

A cut finite element method for a model of pressure in fractured media

We develop a robust cut finite element method for a model of diffusion in fractured media consisting of a bulk domain with embedded cracks. The crack has its own pressure field and can cut through the bulk mesh in a very general fashion. Starting from a common background bulk mesh, that covers the domain, finite element spaces are constructed for the interface and bulk subdomains leading to efficient computations of the coupling terms. The crack pressure field also uses the bulk mesh for its representation. The interface conditions are a generalized form of conditions of Robin type previously considered in the literature which allows the modeling of a range of flow regimes across the fracture. The method is robust in the following way: 1. Stability of the formulation in the full range of parameter choices; and 2. Not sensitive to the location of the interface in the background mesh. We derive an optimal order a priori error estimate and present illustrating numerical examples.

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arXiv2020

Application of a minimal compatible element to incompressible and nearly incompressible continuum mechanics

In this note we will explore some applications of the recently constructed piecewise affine, $H^1$-conforming element that fits in a discrete de Rham complex [Christiansen and Hu, Generalized finite element systems for smooth differential forms and Stokes' problem. Numer. Math. 140 (2018)]. In particular we show how the element leads to locking free methods for incompressible elasticity and viscosity robust methods for the Brinkman model.

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arXiv2019

Cut Bogner-Fox-Schmit Elements for Plates

We present and analyze a method for thin plates based on cut Bogner-Fox-Schmit elements, which are $C^1$ elements obtained by taking tensor products of Hermite splines. The formulation is based on Nitsche's method for weak enforcement of essential boundary conditions together with addition of certain stabilization terms that enable us to establish coercivity and stability of the resulting system of linear equations. We also take geometric approximation of the boundary into account and we focus our presentation on the simply supported boundary conditions which is the most sensitive case for geometric approximation of the boundary.

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arXiv2019

Comparison of mechanical conditions in a lower leg model with 5 or 6 tissue types while exposed to prosthetic sockets applying finite element analysis

Lower limb amputees often suffer skin and tissue problems from using their prosthesis which is a challenging biomechanical problem. The finite element method (FEM) has previously been applied to analyse internal mechanical conditions of the leg at prosthesis use. However, the representation of soft tissue was simplified to few layers and tissue types. The effects of such a simplification of human tissue is still unclear and the results from simplified models may be misleading. Thus, comparisons of the effects of using five versus six tissue types were performed on a transtibial cross section model exposed to three different socket designs. Skin, fat, vessels and bones were defined separately while muscle and fascia tissues were separate or merged. Nonlinear behaviour and friction between socket and skin were considered in the simulations. Contact forces as well as internal stresses and strains of each tissue type differed in both magnitude and maxima site for each material set within and between the different prosthetic socket conditions. Relative changes of stresses and strains by several hundred percent were found when fascia and muscle material properties were merged compared to when they were modelled separately. Thus, the level of tissue detail needs to be considered when creating limb models and interpreting results of related FEM simulations. Keywords: mechanical condition, finite element, soft tissue, material property, contact, prosthesis

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arXiv2019

Dirichlet Boundary Value Correction using Lagrange Multipliers

We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to the Taylor series expansion approach used by Bramble, Dupont and Tomée [Math. Comp., 26:869--879, 1972] in the context of Nitsche's method and, in the case of inf-sup stable multiplier methods, prove a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.

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arXiv2019

Cut Topology Optimization for Linear Elasticity with Coupling to Parametric Nondesign Domain Regions

We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represents parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche's method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero.

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arXiv2018

Analysis of Finite Element Methods for Vector Laplacians on Surfaces

We develop a finite element method for the vector Laplacian based on the covariant derivative of tangential vector fields on surfaces embedded in $\mathbb{R}^3$. 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 which describe a $\mathbb{R}^3$ vector field on the surface and the tangent condition is weakly enforced using a penalization term. We derive error estimates that take the approximation of both the geometry of the surface and the solution to the partial differential equation into account. 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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arXiv2018

Hybridized CutFEM for Elliptic Interface Problems

We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples.

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arXiv2018

A Stable Cut Finite Element Method for Partial Differential Equations on Surfaces: The Helmholtz-Beltrami Operator

We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition $h k < C$, where $h$ denotes the mesh size, $k$ the wave number and $C$ a constant depending mainly on the surface curvature $κ$, but not on the surface/mesh intersection. Optimal error estimates in the $H^1$ and $L^2$-norms follow.

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arXiv2018

A Stabilized Cut Streamline Diffusion Finite Element Method for Convection-Diffusion Problems on Surfaces

We develop a stabilized cut finite element method for the stationary convection diffusion problem on a surface embedded in ${\mathbb{R}}^d$. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order $O(h^{-1})$ for a specific choice of method parameters. Numerical example supporting our theoretical results are also included.

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arXiv2018

Cut Finite Elements for Convection in Fractured Domains

We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain which is a union of manifolds of different dimensions such that a $d$ dimensional component always resides on the boundary of a $d+1$ dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem can be formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is based on using a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.

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arXiv2018

A Cut Finite Element Method for Elliptic Bulk Problems with Embedded Surfaces

We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractured. In particular the Laplace-Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples.

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arXiv2018

Modified midpoint integration rule for the trilinear element in large deformation elasticity

In this paper we suggest two modified one-point Gauss integration rules for the Q1 bi- or trilinear element. The modifications both stabilize the hourglass modes of the one-point rule, and one of them is accurate also on severely distorted elements. We investigate the performance of the integration rules for the hexahedron element, and combine standard one-point integration of the volumetric terms with the modified rules for the isochoric terms to handle near incompressible situations.

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arXiv2018

A cut finite element method with boundary value correction for the incompressible Stokes' equations

We design a cut finite element method for the incompressible Stokes equations on curved domains. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche's method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman, Hansbo, Larson, 'A cut finite element method with boundary value correction', Math. Comp. 87(310):633--657, 2018.

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arXiv2017

A Simple Finite Element Method for Elliptic Bulk Problems with Embedded Surfaces

In this paper we develop a simple finite element method for simulation of embedded layers of high permeability in a matrix of lower permeability using a basic model of Darcy flow in embedded cracks. The cracks are allowed to cut through the mesh in arbitrary fashion and we take the flow in the crack into account by superposition. The fact that we use continuous elements leads to suboptimal convergence due to the loss of regularity across the crack. We therefore refine the mesh in the vicinity of the crack in order to recover optimal order convergence in terms of the global mesh parameter. The proper degree of refinement is determined based on an a priori error estimate and can thus be performed before the actual finite element computation is started. Numerical examples showing this effect and confirming the theoretical results are provided. The approach is easy to implement and beneficial for rapid assessment of the effect of crack orientation and may for example be used in an optimization loop.

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