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Larson, Mats G.

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, Erik

### Hansbo, Peter

### Larson, Mats G.

### Massing, André

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_0_j_idt195_some",{id:"formSmash:j_idt191:0:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_0_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_0_j_idt195_otherAuthors",{id:"formSmash:j_idt191:0:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_0_j_idt195_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) Published
##### Abstract [en]

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

Elsevier, 2020
##### Keywords

Helmholtz-Beltrami, TraceFEM, stabilization
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-169084 (URN)10.1016/j.cma.2019.112803 (DOI)000515542500013 ()
#####

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Available from: 2020-03-23 Created: 2020-03-23 Last updated: 2020-03-23Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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 kappa < C, where h denotes the mesh size, kappa the wave number and C a constant depending mainly on the surface curvature kappa, but not on the surface/mesh intersection. Optimal error estimates in the H-1 and L-2-norms follow.

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_idt191_1_j_idt195_some",{id:"formSmash:j_idt191:1:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_1_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_1_j_idt195_otherAuthors",{id:"formSmash:j_idt191:1:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_1_j_idt195_otherAuthors",multiple:true}); 2020 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 60, no 1, p. 235-260Article in journal (Refereed) Published
##### Abstract [en]

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

Springer, 2020
##### Keywords

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

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-169475 (URN)10.1007/s10543-019-00773-4 (DOI)000519379000009 ()
#####

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Available from: 2020-04-02 Created: 2020-04-02 Last updated: 2020-04-02Bibliographically approved

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.

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.

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.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Samvin, David

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_2_j_idt195_some",{id:"formSmash:j_idt191:2:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_2_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_2_j_idt195_otherAuthors",{id:"formSmash:j_idt191:2:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_2_j_idt195_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 Berlin/Heidelberg, 2019
##### Keywords

Finite element, Unfitted, Embedded, Fractures
##### National Category

Computer Sciences Probability Theory and Statistics
##### Identifiers

urn:nbn:se:umu:diva-158762 (URN)10.1007/s13137-019-0120-z (DOI)000463142200001 ()30873244 (PubMedID)
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Available from: 2019-05-13 Created: 2019-05-13 Last updated: 2020-03-10Bibliographically approved

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 new least squares stabilized Nitsche method for cut isogeometric analysis### Elfverson, Daniel

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Larson, Mats G.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Larsson, Karl

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_some",{id:"formSmash:j_idt191:3:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_3_j_idt195_otherAuthors",{id:"formSmash:j_idt191:3:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_3_j_idt195_otherAuthors",multiple:true}); 2019 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 349, p. 1-16Article in journal (Refereed) Published
##### Abstract [en]

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

Elsevier, 2019
##### Keywords

Fictitious domain methods, Nitsche’s method, Least squares stabilization, Isogeometric analysis
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-156840 (URN)10.1016/j.cma.2019.02.011 (DOI)2-s2.0-85062154279 (Scopus ID)
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##### Funder

Swedish Research Council, 2013-4708Swedish Research Council, 2017-03911Swedish Foundation for Strategic Research , AM13-0029eSSENCE - An eScience Collaboration
Available from: 2019-03-01 Created: 2019-03-01 Last updated: 2019-06-13Bibliographically approved

We derive a new stabilized symmetric Nitsche method for enforcement of Dirichlet boundary conditions for elliptic problems of second order in cut isogeometric analysis (CutIGA). We consider *C*^{1} splines and stabilize the standard Nitsche method by adding a certain elementwise least squares terms in the vicinity of the Dirichlet boundary and an additional term on the boundary which involves the tangential gradient. We show coercivity with respect to the energy norm for functions in *H*^{2}(Ω) and optimal order a priori error estimates in the energy and *L*^{2} norms. To obtain a well posed linear system of equations we combine our formulation with basis function removal which essentially eliminates basis functions with sufficiently small intersection with Ω. The upshot of the formulation is that only elementwise stabilization is added in contrast to standard procedures based on ghost penalty and related techniques and that the stabilization is consistent. In our numerical experiments we see that the method works remarkably well in even extreme cut situations using a Nitsche parameter of moderate size.

Open this publication in new window or tab >>A simple embedded discrete fracture-matrix model for a coupled flow and transport problem in porous media### Odsæter, Lars H.

### Kvamsdal, Trond

### Larson, Mats G.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_4_j_idt195_some",{id:"formSmash:j_idt191:4:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_4_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_4_j_idt195_otherAuthors",{id:"formSmash:j_idt191:4:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_4_j_idt195_otherAuthors",multiple:true}); 2019 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 343, p. 572-601Article in journal (Refereed) Published
##### Abstract [en]

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

Elsevier, 2019
##### Keywords

Discrete fracture-matrix model, Embedded interface, Finite element method, Finite volume method, Porous media flow
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-153110 (URN)10.1016/j.cma.2018.09.003 (DOI)000447411100026 ()
#####

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Available from: 2018-11-12 Created: 2018-11-12 Last updated: 2018-11-12Bibliographically approved

Accurate simulation of fluid flow and transport in fractured porous media is a key challenge in subsurface reservoir engineering. Due to the high ratio between its length and width, fractures can be modeled as lower dimensional interfaces embedded in the porous rock. We apply a recently developed embedded finite element method (EFEM) for the Darcy problem. This method allows for general fracture geometry, and the fractures may cut the finite element mesh arbitrarily. We present here a velocity model for EFEM and couple the Darcy problem to a transport problem for a passive solute. The main novelties of this work are a locally conservative velocity approximation derived from the EFEM solution, and the development of a lowest order upwind finite volume method for the transport problem. This numerical model is compatible with EFEM in the sense that the same computational mesh may be applied, so that we retain the same flexibility with respect to fracture geometry and meshing. Hence, our coupled solution strategy represents a simple approach in terms of formulation, implementation and meshing. We demonstrate our model by some numerical examples on both synthetic and realistic problems, including a benchmark study for single-phase flow. Despite the simplicity of the method, the results are promising.

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.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_5_j_idt195_some",{id:"formSmash:j_idt191:5:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_5_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_5_j_idt195_otherAuthors",{id:"formSmash:j_idt191:5:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_5_j_idt195_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

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

Computer Sciences Probability Theory and Statistics
##### Identifiers

urn:nbn:se:umu:diva-157594 (URN)10.1007/s10596-018-9792-y (DOI)000459423400010 ()
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Available from: 2019-03-28 Created: 2019-03-28 Last updated: 2019-03-28Bibliographically approved

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.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_6_j_idt195_some",{id:"formSmash:j_idt191:6:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_6_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_6_j_idt195_otherAuthors",{id:"formSmash:j_idt191:6:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_6_j_idt195_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]

##### Keywords

Signorini problem, obstacle problem, finite element method, Lagrange mutlipliers, augmented grangian, error estimates
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:umu:diva-158369 (URN)10.1051/m2an/2018047 (DOI)000464277200001 ()
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##### Funder

Swedish Research Council, 2011-4992Swedish Research Council, 2013-4708
Available from: 2019-04-26 Created: 2019-04-26 Last updated: 2019-04-26Bibliographically approved

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.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Massing, André

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_7_j_idt195_some",{id:"formSmash:j_idt191:7:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_7_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_7_j_idt195_otherAuthors",{id:"formSmash:j_idt191:7:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_7_j_idt195_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

Surface PDE, Laplace-Beltrami operator, cut finite element method, stabilization, condition number, a priori error estimates, arbitrary codimension
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-162515 (URN)10.1051/m2an/2018038 (DOI)000457984700005 ()
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Available from: 2019-08-21 Created: 2019-08-21 Last updated: 2020-03-20Bibliographically approved

We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in R-d 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 R-3.

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

### Peter, Hansbo

### Larson, Mats G.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Larsson, Karl

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_some",{id:"formSmash:j_idt191:8:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_8_j_idt195_otherAuthors",{id:"formSmash:j_idt191:8:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_8_j_idt195_otherAuthors",multiple:true}); 2019 (English)In: Computers & Fluids, ISSN 0045-7930, E-ISSN 1879-0747, Vol. 179, p. 726-734Article in journal (Refereed) Published
##### Abstract [en]

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

Elsevier, 2019
##### Keywords

Convection problems, Fractured domains, Mixed-dimensional domains, Galerkin least squares, A priori error estimates
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-158254 (URN)10.1016/j.compfluid.2018.07.022 (DOI)000467514000053 ()2-s2.0-85052134188 (Scopus ID)
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##### Funder

Swedish Research Council, 2013-4708Swedish Research Council, 2017-03911Swedish Research Council, 2018-05262Swedish Foundation for Strategic Research , AM13-0029eSSENCE - An eScience Collaboration, -
Available from: 2019-04-17 Created: 2019-04-17 Last updated: 2019-06-18Bibliographically approved

University College London, UK, Department of Mathematics.

Jönköping University, School of Engineering, JTH, Product Development.

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

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.### Hansbo, Peter

### Larson, Mats G.

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_9_j_idt195_some",{id:"formSmash:j_idt191:9:j_idt195:some",widgetVar:"widget_formSmash_j_idt191_9_j_idt195_some",multiple:true}); ### Larsson, Karl

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt191_9_j_idt195_otherAuthors",{id:"formSmash:j_idt191:9:j_idt195:otherAuthors",widgetVar:"widget_formSmash_j_idt191_9_j_idt195_otherAuthors",multiple:true}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt191_9_j_idt195_j_idt209",{id:"formSmash:j_idt191:9:j_idt195:j_idt209",widgetVar:"widget_formSmash_j_idt191_9_j_idt195_j_idt209",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

Material distribution topology optimization, Design and nondesign domain regions, Cut finite element method
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-157679 (URN)10.1016/j.cma.2019.03.016 (DOI)000468163500019 ()
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##### Funder

Swedish Research Council, 2013-4708Swedish Research Council, 2017-03911Swedish Research Council, 2018-05262Swedish Foundation for Strategic Research , AM13-0029eSSENCE - An eScience Collaboration, -
Available from: 2019-03-29 Created: 2019-03-29 Last updated: 2019-06-11Bibliographically approved

University College London, UK, Department of Mathematics.

Jönköping University, School of Engineering, JTH, Product Development.

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.