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Hansbo, Peterorcid.org/0000-0001-7352-1550

Open this publication in new window or tab >>A stabilized cut streamline diffusion finite element method for convection–diffusion problems on surfaces### Burman, Erik

### Hansbo, Peter

### Larson, Mats G.

### Massing, André

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_some",{id:"formSmash:j_idt184:0:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_some",multiple:true}); ### Zahedi, Sara

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_0_j_idt188_otherAuthors",{id:"formSmash:j_idt184:0:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_0_j_idt188_j_idt202",{id:"formSmash:j_idt184:0:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_0_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2020 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 358, article id 112645Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2020
##### Keywords

Continuous interior penalty, Convection–diffusion–reaction, Cut finite element method, PDEs on surfaces, Streamline diffusion, Diffusion, Mesh generation, Number theory, Piecewise linear techniques, Stabilization, Stiffness matrix, Interior penalties, Piecewise linear approximations, Priori error estimate, Streamline diffusion methods, Streamline-diffusion finite element methods, Tetrahedral elements, Finite element method
##### National Category

Materials Engineering Mathematics
##### Identifiers

urn:nbn:se:hj:diva-46549 (URN)10.1016/j.cma.2019.112645 (DOI)000496915700036 ()2-s2.0-85072756632 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
#####

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##### Funder

Swedish Research Council, 2011-4992, 2013-4708Swedish Foundation for Strategic Research , AM13-0029,PH
Available from: 2019-10-14 Created: 2019-10-14 Last updated: 2019-12-05Bibliographically approved

Department of Mathematics, University College London, London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.

Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.

Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden.

We develop a stabilized cut finite element method for the stationary convection–diffusion problem on a surface embedded in Rd. 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 for a specific choice of method parameters. Numerical examples supporting our theoretical results are also included.

Open this publication in new window or tab >>A stable cut finite element method for partial differential equations on surfaces: The Helmholtz–Beltrami operator### Burman, E.

### Hansbo, Peter

### Larson, M. G.

### Massing, A.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2020 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 362, article id 112803Article in journal (Refereed) Epub ahead of print
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2020
##### Keywords

Helmholtz-Beltrami, stabilization, TraceFEM, Finite element method, Galerkin methods, Mesh generation, Piecewise linear techniques, Beltrami, Discrete formulations, Galerkin Least Squares, Optimal error estimate, Surface curvatures, Three dimensional space, Two-dimensional surface, Least squares approximations
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:hj:diva-47450 (URN)10.1016/j.cma.2019.112803 (DOI)2-s2.0-85077774353 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
#####

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Available from: 2020-01-20 Created: 2020-01-20 Last updated: 2020-01-20

Department of Mathematics, University College London, London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.

Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.

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 hk<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 H1 and L2-norms follow.

Open this publication in new window or tab >>A cut finite element method for elliptic bulk problems with embedded surfaces### Burman, Erik

### Hansbo, Peter

### Larson, Mats G.

### Samvin, David

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_some",{id:"formSmash:j_idt184:2:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_2_j_idt188_otherAuthors",{id:"formSmash:j_idt184:2:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_2_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: GEM - International Journal on Geomathematics, ISSN 1869-2672, E-ISSN 1869-2680, Vol. 10, no 1, article id 10Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2019
##### Keywords

Embedded, Finite element, Fractures, Unfitted, Fracture, Porous materials, Domain geometry, Embedded surfaces, Fractured porous media, Normal component, Optimal order error estimates, Regularity assumption, Finite element method
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:hj:diva-43354 (URN)10.1007/s13137-019-0120-z (DOI)000463142200001 ()30873244 (PubMedID)2-s2.0-85061086676 (Scopus ID)HOA JTH 2019;JTHMaterialIS (Local ID)HOA JTH 2019;JTHMaterialIS (Archive number)HOA JTH 2019;JTHMaterialIS (OAI)
#####

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Available from: 2019-03-19 Created: 2019-03-19 Last updated: 2019-05-08Bibliographically approved

Mathematics, University College London, London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

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 fractures. 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.

Open this publication in new window or tab >>A cut finite element method with boundary value correction for the incompressible Stokes equations### Burman, Erik N.

### Hansbo, Peter

### Larson, Mats G.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Numerical mathematics and advanced applications ENUMATH 2017, Cham: Springer, 2019, Vol. 126, p. 183-192Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Cham: Springer, 2019
##### Series

Lecture Notes in Computational Science and Engineering, ISSN 14397358 ; 126
##### Keywords

Boundary conditions, Navier Stokes equations, Computational mesh, Correction techniques, Curved boundary, Dirichlet boundary condition, Discrete boundaries, Discrete domains, Incompressible Stokes equation, Piecewise affines, Finite element method
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:hj:diva-42799 (URN)10.1007/978-3-319-96415-7_15 (DOI)2-s2.0-85060026956 (Scopus ID)9783319964140 (ISBN)9783319964157 (ISBN)
##### Conference

European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017, Voss, Norway, 25 - 29 September 2017
#####

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Available from: 2019-02-01 Created: 2019-02-01 Last updated: 2019-02-15Bibliographically approved

UCL, Department of Mathematics, London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Umeå Universitet, Department of Mathematics and Mathematical Statistics, Umeå, Sweden.

We design a cut finite element method for the incompressible Stokes equations on domains with curved boundary. 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 et al. (Math Comput 87(310):633–657, 2018).

Open this publication in new window or tab >>A simple finite element method for elliptic bulk problems with embedded surfaces### Burman, Erik

### Hansbo, Peter

### Larson, Mats G.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Computational Geosciences, ISSN 1420-0597, E-ISSN 1573-1499, Vol. 23, no 1, p. 189-199Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2019
##### Keywords

Cut finite element methods, Darcy equation, Embedded layer, Fracture
##### National Category

Computer Engineering
##### Identifiers

urn:nbn:se:hj:diva-42144 (URN)10.1007/s10596-018-9792-y (DOI)000459423400010 ()2-s2.0-85056316370 (Scopus ID)HOA JTH 2019;JTHProduktutvecklingIS (Local ID)HOA JTH 2019;JTHProduktutvecklingIS (Archive number)HOA JTH 2019;JTHProduktutvecklingIS (OAI)
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Available from: 2018-11-26 Created: 2018-11-26 Last updated: 2019-03-14Bibliographically approved

Department of Mathematics, University College London, London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.

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.

Open this publication in new window or tab >>Augmented Lagrangian finite element methods for contact problems### Burman, Erik

### Hansbo, Peter

### Larson, Mats G.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 53, no 1, p. 173-195Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

EDP Sciences, 2019
##### Keywords

Augmented Lagrangian, Error estimates, Finite element method, Lagrange mutlipliers, Obstacle problem, Signorini problem, Constrained optimization, Augmented Lagrangians, Lagrange, Obstacle problems, Lagrange multipliers
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:hj:diva-43538 (URN)10.1051/m2an/2018047 (DOI)000464277200001 ()2-s2.0-85064252246 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
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Available from: 2019-04-25 Created: 2019-04-25 Last updated: 2019-04-25Bibliographically approved

Department of Mathematics, University College London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.

We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution.

Open this publication in new window or tab >>Cut finite element methods for partial differential equations on embedded manifolds of arbitrary codimensions### Burman, Erik

### Hansbo, Peter

### Larson, Mats G.

### Massing, André

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, E-ISSN 1290-3841, Vol. 52, no 6, p. 2247-2282Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

EDP Sciences, 2019
##### Keywords

A priori error estimates, Arbitrary codimension, Condition number, Cut finite element method, Laplace-Beltrami operator, Stabilization, Surface PDE, Embeddings, Laplace transforms, Mesh generation, Number theory, Piecewise linear techniques, Stiffness matrix, A-priori estimates, Codimension, Condition number of the stiffness matrix, Condition numbers, Numerical results, Priori error estimate, Theoretical framework, Finite element method
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:hj:diva-43224 (URN)10.1051/m2an/2018038 (DOI)000457984700005 ()2-s2.0-85052126397 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
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Available from: 2019-03-01 Created: 2019-03-01 Last updated: 2019-03-01Bibliographically approved

Department of Mathematics, University College London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.

Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in Rd of arbitrary codimension. The method is based on using continuous piecewise linears on a background mesh in the embedding space for approximation together with a stabilizing form that ensures that the resulting problem is stable. The discrete manifold is represented using a triangulation which does not match the background mesh and does not need to be shape-regular, which includes level set descriptions of codimension one manifolds and the non-matching embedding of independently triangulated manifolds as special cases. We identify abstract key assumptions on the stabilizing form which allow us to prove a bound on the condition number of the stiffness matrix and optimal order a priori estimates. The key assumptions are verified for three different realizations of the stabilizing form including a novel stabilization approach based on penalizing the surface normal gradient on the background mesh. Finally, we present numerical results illustrating our results for a curve and a surface embedded in R3.

Open this publication in new window or tab >>Cut finite elements for convection in fractured domains### Burman, Erik

### Hansbo, Peter

### Larson, Mats G.

### Larsson, Karl

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 179, p. 728-736Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2019
##### Keywords

A priori error estimates, Convection problems, Fractured domains, Galerkin least squares, Mixed-dimensional domains, Fracture, Mesh generation, Piecewise linear techniques, Porous materials, Convection problem, Coupling condition, Directional derivative, Divergence operators, Modeling porous medias, Piecewise linear, Priori error estimate, Finite element method
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:hj:diva-41509 (URN)10.1016/j.compfluid.2018.07.022 (DOI)000467514000053 ()2-s2.0-85052134188 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
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Available from: 2018-09-19 Created: 2018-09-19 Last updated: 2020-01-20Bibliographically approved

Department of Mathematics, University College London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.

Department of Mathematics and Mathematical Statistics, Umeå University, Sweden.

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 is formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is posed on 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.

Open this publication in new window or tab >>Cut topology optimization for linear elasticity with coupling to parametric nondesign domain regions### Burman, Erik

### Elfverson, Daniel

### Hansbo, Peter

### Larson, Mats G.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); ### Larsson, Karl

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_8_j_idt188_j_idt202",{id:"formSmash:j_idt184:8:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2019 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 350, p. 462-479Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Elsevier, 2019
##### Keywords

Cut finite element method, Design and nondesign domain regions, Material distribution topology optimization, Elasticity, Mesh generation, Stabilization, Topology, Complicated geometry, Density parameters, Dirichlet condition, Interface conditions, Isogeometric analysis, Linear elasticity, Material distribution, Topology Optimization Method, Finite element method
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:hj:diva-43449 (URN)10.1016/j.cma.2019.03.016 (DOI)000468163500019 ()2-s2.0-85063489165 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
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Available from: 2019-04-10 Created: 2019-04-10 Last updated: 2019-06-07Bibliographically approved

University College London, London, United Kingdom.

Umeå University, Umeå, Sweden.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Umeå University, Umeå, Sweden.

Umeå University, Umeå, Sweden.

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 represent 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.

Open this publication in new window or tab >>Dirichlet boundary value correction using Lagrange multipliers### Burman, Erik

### Hansbo, Peter

### Larson, Mats G.

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2019 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125Article in journal (Refereed) Epub ahead of print
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2019
##### Keywords

Boundary value correction, Dirichlet boundary conditions, Lagrange multiplier
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:hj:diva-46297 (URN)10.1007/s10543-019-00773-4 (DOI)2-s2.0-85072217688 (Scopus ID);JTHMaterialIS (Local ID);JTHMaterialIS (Archive number);JTHMaterialIS (OAI)
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Available from: 2019-09-23 Created: 2019-09-23 Last updated: 2019-09-23

Department of Mathematics, University College London, London, United Kingdom.

Jönköping University, School of Engineering, JTH, Materials and Manufacturing.

Department of Mathematics and Mathematical Statistics, Umeå University, Umeå, Sweden.

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 a Taylor series expansion approach previously used 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.