209 research outputs found
Reliable and efficient a posteriori error estimation for adaptive IGA boundary element methods for weakly-singular integral equations
We consider the Galerkin boundary element method (BEM) for weakly-singular
integral equations of the first-kind in 2D. We analyze some residual-type a
posteriori error estimator which provides a lower as well as an upper bound for
the unknown Galerkin BEM error. The required assumptions are weak and allow for
piecewise smooth parametrizations of the boundary, local mesh-refinement, and
related standard piecewise polynomials as well as NURBS. In particular, our
analysis gives a first contribution to adaptive BEM in the frame of
isogeometric analysis (IGABEM), for which we formulate an adaptive algorithm
which steers the local mesh-refinement and the multiplicity of the knots.
Numerical experiments underline the theoretical findings and show that the
proposed adaptive strategy leads to optimal convergence
Residual-based error estimation and adaptivity for stabilized immersed isogeometric analysis using truncated hierarchical B-splines
We propose an adaptive mesh refinement strategy for immersed isogeometric
analysis, with application to steady heat conduction and viscous flow problems.
The proposed strategy is based on residual-based error estimation, which has
been tailored to the immersed setting by the incorporation of appropriately
scaled stabilization and boundary terms. Element-wise error indicators are
elaborated for the Laplace and Stokes problems, and a THB-spline-based local
mesh refinement strategy is proposed. The error estimation .and adaptivity
procedure is applied to a series of benchmark problems, demonstrating the
suitability of the technique for a range of smooth and non-smooth problems. The
adaptivity strategy is also integrated in a scan-based analysis workflow,
capable of generating reliable, error-controlled, results from scan data,
without the need for extensive user interactions or interventions.Comment: Submitted to Journal of Mechanic
Analysis-suitable adaptive T-mesh refinement with linear complexity
We present an efficient adaptive refinement procedure that preserves
analysis-suitability of the T-mesh, this is, the linear independence of the
T-spline blending functions. We prove analysis-suitability of the overlays and
boundedness of their cardinalities, nestedness of the generated T-spline
spaces, and linear computational complexity of the refinement procedure in
terms of the number of marked and generated mesh elements.Comment: We now account for T-splines of arbitrary polynomial degree. We
replaced the proof of Dual-Compatibility by a proof of Analysis-suitability,
added a section where we address nestedness of the corresponding T-spline
spaces, and removed the section on finite overlap the spline supports. 24
pages, 9 Figure
Isogeometric analysis based on Geometry Independent Field approximaTion (GIFT) and Polynomial Splines over Hierarchical T-meshes
This thesis addresses an adaptive higher-order method based on a Geometry Independent Field approximatTion(GIFT) of polynomial/rationals plines over hierarchical T-meshes(PHT/RHT-splines).
In isogeometric analysis, basis functions used for constructing geometric models in computer-aided design(CAD) are also employed to discretize the partial differential equations(PDEs) for numerical analysis. Non-uniform rational B-Splines(NURBS) are the most commonly used basis functions in CAD. However, they may not be ideal for numerical analysis where local refinement is required.
The alternative method GIFT deploys different splines for geometry and numerical analysis. NURBS are utilized for the geometry representation, while for the field solution, PHT/RHT-splines are used. PHT-splines not only inherit the useful properties of B-splines and NURBS, but also possess the capabilities of local refinement and hierarchical structure. The smooth basis function properties of PHT-splines make them suitable for analysis purposes. While most problems considered in isogeometric analysis can be solved efficiently when the solution is smooth, many non-trivial problems have rough solutions. For example, this can be caused by the presence of re-entrant corners in the domain. For such problems, a tensor-product basis (as in the case of NURBS) is less suitable for resolving the singularities that appear since refinement propagates throughout the computational domain. Hierarchical bases and local refinement (as in the case of PHT-splines) allow for a more efficient way to resolve these singularities by adding more degrees of freedom where they are necessary. In order to drive the adaptive refinement, an efficient recovery-based error estimator is proposed in this thesis. The estimator produces a recovery solution which is a more accurate approximation than the computed numerical solution. Several two- and three-dimensional numerical investigations with PHT-splines of higher order and continuity prove that the proposed method is capable of obtaining results with higher accuracy, better convergence, fewer degrees of freedom and less computational cost than NURBS for smooth solution problems. The adaptive GIFT method utilizing PHT-splines with the recovery-based error estimator is used for solutions with discontinuities or singularities where adaptive local refinement in particular domains of interest achieves higher accuracy with fewer degrees of freedom. This method also proves that it can handle complicated multi-patch domains for two- and three-dimensional problems outperforming uniform refinement in terms of degrees of freedom and computational cost
Mathematical foundations of adaptive isogeometric analysis
This paper reviews the state of the art and discusses recent developments in
the field of adaptive isogeometric analysis, with special focus on the
mathematical theory. This includes an overview of available spline technologies
for the local resolution of possible singularities as well as the
state-of-the-art formulation of convergence and quasi-optimality of adaptive
algorithms for both the finite element method (FEM) and the boundary element
method (BEM) in the frame of isogeometric analysis (IGA)
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