116 research outputs found
Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT
In this paper, we develop a shape optimization-based algorithm for the
electrical impedance tomography (EIT) problem of determining a piecewise
constant conductivity on a polygonal partition from boundary measurements. The
key tool is to use a distributed shape derivative of a suitable cost functional
with respect to movements of the partition. Numerical simulations showing the
robustness and accuracy of the method are presented for simulated test cases in
two dimensions
Wavelet-Fourier CORSING techniques for multi-dimensional advection-diffusion-reaction equations
We present and analyze a novel wavelet-Fourier technique for the numerical
treatment of multidimensional advection-diffusion-reaction equations based on
the CORSING (COmpRessed SolvING) paradigm. Combining the Petrov-Galerkin
technique with the compressed sensing approach, the proposed method is able to
approximate the largest coefficients of the solution with respect to a
biorthogonal wavelet basis. Namely, we assemble a compressed discretization
based on randomized subsampling of the Fourier test space and we employ sparse
recovery techniques to approximate the solution to the PDE. In this paper, we
provide the first rigorous recovery error bounds and effective recipes for the
implementation of the CORSING technique in the multi-dimensional setting. Our
theoretical analysis relies on new estimates for the local a-coherence, which
measures interferences between wavelet and Fourier basis functions with respect
to the metric induced by the PDE operator. The stability and robustness of the
proposed scheme is shown by numerical illustrations in the one-, two-, and
three-dimensional case
Model adaptation enriched with an anisotropic mesh spacing for nonlinear equations: application to environmental and CFD problems
Goal of this paper is to suitably combine a model with an anisotropic mesh
adaptation for the numerical simulation of nonlinear advection-diffusion-reaction systems and incompressible flows in ecological and environmental applications. Using the reduced-basis method terminology, the proposed approach leads to a noticeable computational saving of the online phase with respect to the resolution of the reference model on nonadapted grids. The search of a suitable adapted model/mesh pair is to be meant, instead, in an offline fashion
Spatial regression models over two-dimensional manifolds
We propose a regression model for data spatially distributed over general two-dimensional Riemannian manifolds. This is a generalized additive model with a roughness penalty term involving a differential operator computed over the non-planar domain. By virtue of a semiparametric framework, the model allows inclusion of space-varying covariate information. Estimation can be performed by conformally parameterizing the non-planar domain and then generalizing existing models for penalized spatial regression over planar domains. The conformal coordinates and the estimation problem are both computed with a finite element approach
Scalable Recovery-based Adaptation on Quadtree Meshes for Advection-Diffusion-Reaction Problems
We propose a mesh adaptation procedure for Cartesian quadtree meshes, to
discretize scalar advection-diffusion-reaction problems. The adaptation process
is driven by a recovery-based a posteriori estimator for the -norm
of the discretization error, based on suitable higher order approximations of
both the solution and the associated gradient. In particular, a metric-based
approach exploits the information furnished by the estimator to iteratively
predict the new adapted mesh. The new mesh adaptation algorithm is successfully
assessed on different configurations, and turns out to perform well also when
dealing with discontinuities in the data as well as in the presence of internal
layers not aligned with the Cartesian directions. A cross-comparison with a
standard estimate--mark--refine approach and with other adaptive strategies
available in the literature shows the remarkable accuracy and parallel
scalability of the proposed approach
A dimension-reduction model for brittle fractures on thin shells with mesh adaptivity
In this paper, we derive a new 2D brittle fracture model for thin shells via dimension reduction, where the admissible displacements are only normal to the shell surface. The main steps include to endow the shell with a small thickness, to express the three-dimensional energy in terms of the variational model of brittle fracture in linear elasticity, and to study the Γ-limit of the functional as the thickness tends to zero.
The numerical discretization is tackled by first approximating the fracture through a phase field, following an Ambrosio–Tortorelli like approach, and then resorting to an alternating minimization procedure, where the irreversibility of the crack propagation is rigorously imposed via an inequality constraint. The minimization is enriched with an anisotropic mesh adaptation driven by an a posteriori error estimator, which allows us to sharply track the whole crack path by optimizing the shape, the size, and the orientation of the mesh elements.
Finally, the overall algorithm is successfully assessed on two Riemannian settings and proves not to bias the crack propagation
Hierarchical Model Reduction Techniques for Flow Modeling in a Parametrized Setting
In this work we focus on two different methods to deal with parametrized
partial differential equations in an efficient and accurate way. Starting from
high fidelity approximations built via the hierarchical model reduction
discretization, we consider two approaches, both based on a projection model
reduction technique. The two methods differ for the algorithm employed during
the construction of the reduced basis. In particular, the former employs the
proper orthogonal decomposition, while the latter relies on a greedy algorithm
according to the certified reduced basis technique. The two approaches are
preliminarily compared on two-dimensional scalar and vector test cases
Anisotropic Finite Element Mesh Adaptation via Higher Dimensional Embedding
In this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1], [2], [3], [4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in R3. In the context of adaptive finite element simulation, the solution (which is an unknown function f : Ω ⊂ d → ) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φf (x):= (x1, …, xd, s f (x1, …, xd), s ▿ f (x1, …, xd))t, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function f itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function f. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φf (x) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial differential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG – a metric-based adaptive mesh generator. The errors measured in the L2 norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG
A Dimension-Reduction Model for Brittle Fractures on Thin Shells with Mesh Adaptivity
In this paper we derive a new two-dimensional brittle fracture model for thin
shells via dimension reduction, where the admissible displacements are only
normal to the shell surface. The main steps include to endow the shell with a
small thickness, to express the three-dimensional energy in terms of the
variational model of brittle fracture in linear elasticity, and to study the
-limit of the functional as the thickness tends to zero. The numerical
discretization is tackled by first approximating the fracture through a phase
field, following an Ambrosio-Tortorelli like approach, and then resorting to an
alternating minimization procedure, where the irreversibility of the crack
propagation is rigorously imposed via an inequality constraint. The
minimization is enriched with an anisotropic mesh adaptation driven by an a
posteriori error estimator, which allows us to sharply track the whole crack
path by optimizing the shape, the size, and the orientation of the mesh
elements. Finally, the overall algorithm is successfully assessed on two
Riemannian settings and proves not to bias the crack propagation
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