1,418 research outputs found
A degenerating PDE system for phase transitions and damage
In this paper, we analyze a PDE system arising in the modeling of phase
transition and damage phenomena in thermoviscoelastic materials. The resulting
evolution equations in the unknowns \theta (absolute temperature), u
(displacement), and \chi (phase/damage parameter) are strongly nonlinearly
coupled. Moreover, the momentum equation for u contains \chi-dependent elliptic
operators, which degenerate at the pure phases (corresponding to the values
\chi=0 and \chi=1), making the whole system degenerate. That is why, we have to
resort to a suitable weak solvability notion for the analysis of the problem:
it consists of the weak formulations of the heat and momentum equation, and,
for the phase/damage parameter \chi, of a generalization of the principle of
virtual powers, partially mutuated from the theory of rate-independent damage
processes. To prove an existence result for this weak formulation, an
approximating problem is introduced, where the elliptic degeneracy of the
displacement equation is ruled out: in the framework of damage models, this
corresponds to allowing for partial damage only. For such an approximate
system, global-in-time existence and well-posedness results are established in
various cases. Then, the passage to the limit to the degenerate system is
performed via suitable variational techniques
Hutchinson Trace Estimation for High-Dimensional and High-Order Physics-Informed Neural Networks
Physics-Informed Neural Networks (PINNs) have proven effective in solving
partial differential equations (PDEs), especially when some data are available
by seamlessly blending data and physics. However, extending PINNs to
high-dimensional and even high-order PDEs encounters significant challenges due
to the computational cost associated with automatic differentiation in the
residual loss. Herein, we address the limitations of PINNs in handling
high-dimensional and high-order PDEs by introducing Hutchinson Trace Estimation
(HTE). Starting with the second-order high-dimensional PDEs ubiquitous in
scientific computing, HTE transforms the calculation of the entire Hessian
matrix into a Hessian vector product (HVP). This approach alleviates the
computational bottleneck via Taylor-mode automatic differentiation and
significantly reduces memory consumption from the Hessian matrix to HVP. We
further showcase HTE's convergence to the original PINN loss and its unbiased
behavior under specific conditions. Comparisons with Stochastic Dimension
Gradient Descent (SDGD) highlight the distinct advantages of HTE, particularly
in scenarios with significant variance among dimensions. We further extend HTE
to higher-order and higher-dimensional PDEs, specifically addressing the
biharmonic equation. By employing tensor-vector products (TVP), HTE efficiently
computes the colossal tensor associated with the fourth-order high-dimensional
biharmonic equation, saving memory and enabling rapid computation. The
effectiveness of HTE is illustrated through experimental setups, demonstrating
comparable convergence rates with SDGD under memory and speed constraints.
Additionally, HTE proves valuable in accelerating the Gradient-Enhanced PINN
(gPINN) version as well as the Biharmonic equation. Overall, HTE opens up a new
capability in scientific machine learning for tackling high-order and
high-dimensional PDEs.Comment: Published in Computer Methods in Applied Mechanics and Engineerin
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
Geometric Surface Processing and Virtual Modeling
In this work we focus on two main topics "Geometric Surface Processing" and "Virtual Modeling". The inspiration and coordination for most of the research work contained in the thesis has been driven by the project New Interactive and Innovative Technologies for CAD (NIIT4CAD), funded by the European Eurostars Programme. NIIT4CAD has the ambitious aim of overcoming the limitations of the traditional approach to surface modeling of current 3D CAD systems by introducing new methodologies and technologies based on subdivision surfaces
in a new virtual modeling framework. These innovations will allow designers and engineers to transform quickly and intuitively an idea of shape in a high-quality geometrical model suited for engineering and manufacturing purposes.
One of the objective of the thesis is indeed the reconstruction and modeling of surfaces, representing arbitrary topology objects, starting from 3D irregular curve networks acquired through an ad-hoc smart-pen device.
The thesis is organized in two main parts: "Geometric Surface Processing" and "Virtual Modeling". During the development of the geometric pipeline in our Virtual Modeling system, we faced many challenges that captured our interest and opened new areas of research and experimentation.
In the first part, we present these theories and some applications to Geometric Surface Processing.
This allowed us to better formalize and give a broader understanding on some of the techniques used in our latest advancements on virtual modeling and surface reconstruction.
The research on both topics led to important results that have been published and presented in articles and conferences of international relevance
The Multigrid Image Transform
A second order partial differential operator is applied to an image function.
To this end we consider both the Laplacian and a more general elliptic operator.
By using a multigrid operator known from the so-called approximation property, we derive a multiresolution decomposition of the image without blurring of edges at
coarser levels. We investigate both a linear and a nonlinear variant and compare to some established methods
A Radial Basis Function (RBF) Compact Finite Difference (FD) Scheme for Reaction-Diffusion Equations on Surfaces
We present a new high-order, local meshfree method for numerically solving reaction diffusion equations on smooth surfaces of codimension 1 embedded in â„ťd. The novelty of the method is in the approximation of the Laplace-Beltrami operator for a given surface using Hermite radial basis function (RBF) interpolation over local node sets on the surface. This leads to compact (or implicit) RBF generated finite difference (RBF-FD) formulas for the Laplace-Beltrami operator, which gives rise to sparse differentiation matrices. The method only requires a set of (scattered) nodes on the surface and an approximation to the surface normal vectors at these nodes. Additionally, the method is based on Cartesian coordinates and thus does not suffer from any coordinate singularities. We also present an algorithm for selecting the nodes used to construct the compact RBF-FD formulas that can guarantee the resulting differentiation matrices have desirable stability properties. The improved accuracy and computational cost that can be achieved with this method over the standard (explicit) RBF-FD method are demonstrated with a series of numerical examples. We also illustrate the flexibility and general applicability of the method by solving two different reaction-diffusion equations on surfaces that are defined implicitly and only by point clouds
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