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Elfverson, Daniel

Open this publication in new window or tab >>A new least squares stabilized Nitsche method for cut isogeometric analysis### Elfverson, Daniel

### Larson, Mats G.

### Larsson, Karl

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}); 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}); 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

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.

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

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 >>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_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); ### Larsson, Karl

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.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}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_1_j_idt188_j_idt202",{id:"formSmash:j_idt184:1:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_1_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

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.

Open this publication in new window or tab >>CutIGA with basis function removal### 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_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}); 2018 (English)In: Advanced Modeling and Simulation in Engineering Sciences, ISSN 2213-7467, Vol. 5, no 6Article in journal (Refereed) Published
##### Abstract [en]

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

Springer, 2018
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-145669 (URN)10.1186/s40323-018-0099-2 (DOI)
#####

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Available from: 2018-03-13 Created: 2018-03-13 Last updated: 2018-06-09Bibliographically approved

We consider a cut isogeometric method, where the boundary of the domain is allowed to cut through the background mesh in an arbitrary fashion for a second order elliptic model problem. In order to stabilize the method on the cut boundary we remove basis functions which have small intersection with the computational domain. We determine criteria on the intersection which guarantee that the order of convergence in the energy norm is not affected by the removal. The higher order regularity of the B-spline basis functions leads to improved bounds compared to standard Lagrange elements.

Open this publication in new window or tab >>Shape optimization using the cut finite element method### Burman, Erik

### Elfverson, Daniel

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

### Larson, Mats

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}); ### Larsson, Karl

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}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_3_j_idt188_j_idt202",{id:"formSmash:j_idt184:3:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2018 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 328, p. 242-261Article in journal (Refereed) Published
##### Abstract [en]

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

Lausanne: Elsevier, 2018
##### Keywords

CutFEM, Shape optimization, Level-set, Fictitious domain method, Linear elasticity
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-140281 (URN)10.1016/j.cma.2017.09.005 (DOI)000416218500011 ()
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##### Funder

Swedish Foundation for Strategic Research , AM13-0029Swedish Research Council, 2011-4992Swedish Research Council, 2013-4708eSSENCE - An eScience Collaboration
Available from: 2017-10-04 Created: 2017-10-04 Last updated: 2018-06-09Bibliographically approved

University College London, UK, Department of Mathematics.

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

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK.

We present a cut finite element method for shape optimization in the case of linear elasticity. The elastic domain is defined by a level-set function, and the evolution of the domain is obtained by moving the level-set along a velocity field using a transport equation. The velocity field is the largest decreasing direction of the shape derivative that satisfies a certain regularity requirement and the computation of the shape derivative is based on a volume formulation. Using the cut finite element method no re-meshing is required when updating the domain and we may also use higher order finite element approximations. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements at the boundary, which provides control of the variation of the solution in the vicinity of the boundary. We implement and illustrate the performance of the method in the two-dimensional case, considering both triangular and quadrilateral meshes as well as finite element spaces of different order.

Open this publication in new window or tab >>A cut finite element method for the Bernoulli free boundary value problem### 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_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); ### Larsson, Karl

Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.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}); Show others...PrimeFaces.cw("SelectBooleanButton","widget_formSmash_j_idt184_4_j_idt188_j_idt202",{id:"formSmash:j_idt184:4:j_idt188:j_idt202",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_j_idt202",onLabel:"Hide others...",offLabel:"Show others..."}); 2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 317, p. 598-618Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Free boundary value problem, CutFEM, Shape optimization, Level set, Fictitious domain method
##### National Category

Computational Mathematics
##### Identifiers

urn:nbn:se:umu:diva-133720 (URN)10.1016/j.cma.2016.12.021 (DOI)000398373500024 ()
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Available from: 2017-05-05 Created: 2017-05-05 Last updated: 2018-06-09Bibliographically approved

We develop a cut finite element method for the Bernoulli free boundary problem. The free boundary, represented by an approximate signed distance function on a fixed background mesh, is allowed to intersect elements in an arbitrary fashion. This leads to so called cut elements in the vicinity of the boundary. To obtain a stable method, stabilization terms are added in the vicinity of the cut elements penalizing the gradient jumps across element sides. The stabilization also ensures good conditioning of the resulting discrete system. We develop a method for shape optimization based on moving the distance function along a velocity field which is computed as the H-1 Riesz representation of the shape derivative. We show that the velocity field is the solution to an interface problem and we prove an a priori error estimate of optimal order, given the limited regularity of the velocity field across the interface, for the velocity field in the H-1 norm. Finally, we present illustrating numerical results.

Open this publication in new window or tab >>Multiscale methods for problems with complex geometry### 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.### Målqvist, Axel

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}); 2017 (English)In: Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, E-ISSN 1879-2138, Vol. 321, p. 103-123Article in journal (Refereed) Published
##### Abstract [en]

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

Elsevier, 2017
##### Keywords

multiscale method, complex geometry, a priori error bound
##### National Category

Computational Mathematics Mathematical Analysis
##### Identifiers

urn:nbn:se:umu:diva-137377 (URN)10.1016/j.cma.2017.03.023 (DOI)000402461100006 ()
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Available from: 2017-07-06 Created: 2017-07-06 Last updated: 2018-06-09Bibliographically approved

We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in the energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.

Open this publication in new window or tab >>A Multilevel Monte Carlo Method for Computing Failure Probabilities### Elfverson, Daniel

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

### Målqvist, Axel

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}); 2016 (English)In: SIAM/ASA Journal on Uncertainty Quantification, ISSN 1560-7526, E-ISSN 2166-2525, Vol. 4, no 1, p. 312-330Article in journal (Refereed) Published
##### Abstract [en]

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

Siam publications, 2016
##### Keywords

multilevel Monte Carlo, failure probability, uncertainty quantification, error analysis
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:umu:diva-139175 (URN)10.1137/140984294 (DOI)000407996700014 ()
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Available from: 2017-09-08 Created: 2017-09-08 Last updated: 2018-06-09Bibliographically approved

We propose and analyze a method for computing failure probabilities of systems modeled as numerical deterministic models (e.g., PDEs) with uncertain input data. A failure occurs when a functional of the solution to the model is below (or above) some critical value. By combining recent results on quantile estimation and the multilevel Monte Carlo method, we develop a method that reduces computational cost without loss of accuracy. We show how the computational cost of the method relates to error tolerance of the failure probability. For a wide and common class of problems, the computational cost is asymptotically proportional to solving a single accurate realization of the numerical model, i.e., independent of the number of samples. Significant reductions in computational cost are also observed in numerical experiments.