We investigate discretizations of a geometrically nonlinear elastic Cosserat shell with nonplanar reference configuration originally introduced by B\^irsan, Ghiba, Martin, and Neff in 2019. The shell model includes curvature terms up to order 5 in the shell thickness, which are crucial to reliably simulate high-curvature deformations such as near-folds or creases. The original model is generalized to shells that are not homeomorphic to a subset of $\mathbb{R}^2$. For this, we replace the originally planar parameter domain by an abstract two-dimensional manifold, and verify that the hyperelastic shell energy and three-dimensional reconstruction are invariant under changes of the local coordinate systems. This general approach allows to determine the elastic response for even non-orientable surfaces like the M\"obius strip and the Klein bottle. We discretize the model with a geometric finite element method and, using that geometric finite elements are $H^1$-conforming, prove that the discrete shell model has a solution. Numerical tests then show the general performance and versatility of the model and discretization method.
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Two combined numerical methods for solving time-varying semilinear differential-algebraic equations (DAEs) are obtained. The convergence and correctness of the methods are proved. When constructing the methods, time-varying spectral projectors which can be found numerically are used. This enables to numerically solve the DAE in the original form without additional analytical transformations. To improve the accuracy of the second method, recalculation is used. The developed methods are applicable to the DAEs with the continuous nonlinear part which may not be differentiable in time, and the restrictions of the type of the global Lipschitz condition are not used in the presented theorems on the DAE global solvability and the convergence of the methods. This extends the scope of methods. The fulfillment of the conditions of the global solvability theorem ensures the existence of a unique exact solution on any given time interval, which enables to seek an approximate solution also on any time interval. Numerical examples illustrating the capabilities of the methods and their effectiveness in various situations are provided. To demonstrate this, mathematical models of the dynamics of electrical circuits are considered. It is shown that the results of the theoretical and numerical analyses of these models are consistent.
The Fokker-Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional numerical methods becomes computationally prohibitive. Here, we introduce a fast, scalable, and interpretable method for solving the Fokker-Planck equation which is applicable in higher dimensions. This method approximates the solution as a linear combination of shape-morphing Gaussians with time-dependent means and covariances. These parameters evolve according to the method of reduced-order nonlinear solutions (RONS) which ensures that the approximate solution stays close to the true solution of the PDE for all times. As such, the proposed method approximates the transient dynamics as well as the equilibrium density, when the latter exists. Our approximate solutions can be viewed as an evolution on a finite-dimensional statistical manifold embedded in the space of probability densities. We show that the metric tensor in RONS coincides with the Fisher information matrix on this manifold. We also discuss the interpretation of our method as a shallow neural network with Gaussian activation functions and time-varying parameters. In contrast to existing deep learning methods, our method is interpretable, requires no training, and automatically ensures that the approximate solution satisfies all properties of a probability density.
Hamilton and Moitra (2021) showed that, in certain regimes, it is not possible to accelerate Riemannian gradient descent in the hyperbolic plane if we restrict ourselves to algorithms which make queries in a (large) bounded domain and which receive gradients and function values corrupted by a (small) amount of noise. We show that acceleration remains unachievable for any deterministic algorithm which receives exact gradient and function-value information (unbounded queries, no noise). Our results hold for the classes of strongly and nonstrongly geodesically convex functions, and for a large class of Hadamard manifolds including hyperbolic spaces and the symmetric space $\mathrm{SL}(n) / \mathrm{SO}(n)$ of positive definite $n \times n$ matrices of determinant one. This cements a surprising gap between the complexity of convex optimization and geodesically convex optimization: for hyperbolic spaces, Riemannian gradient descent is optimal on the class of smooth and and strongly geodesically convex functions, in the regime where the condition number scales with the radius of the optimization domain. The key idea for proving the lower bound consists of perturbing the hard functions of Hamilton and Moitra (2021) with sums of bump functions chosen by a resisting oracle.
This paper presents a general, nonlinear isogeometric finite element formulation for rotation-free shells with embedded fibers that captures anisotropy in stretching, shearing, twisting and bending -- both in-plane and out-of-plane. These capabilities allow for the simulation of large sheets of heterogeneous and fibrous materials either with or without matrix, such as textiles, composites, and pantographic structures. The work is a computational extension of our earlier theoretical work [1] that extends existing Kirchhoff-Love shell theory to incorporate the in-plane bending resistance of initially straight or curved fibers. The formulation requires only displacement degrees-of-freedom to capture all mentioned modes of deformation. To this end, isogeometric shape functions are used in order to satisfy the required $C^1$-continuity for bending across element boundaries. The proposed formulation can admit a wide range of material models, such as surface hyperelasticity that does not require any explicit thickness integration. To deal with possible material instability due to fiber compression, a stabilization scheme is added. Several benchmark examples are used to demonstrate the robustness and accuracy of the proposed computational formulation.
This work discusses the correct modeling of the fully nonlinear free surface boundary conditions to be prescribed in water waves flow simulations based on potential flow theory. The main goal of such a discussion is that of identifying a mathematical formulation and a numerical treatment that can be used both to carry out transient simulations, and to compute steady solutions -- for any flow admitting them. In the literature on numerical towing tank in fact, steady and unsteady fully nonlinear potential flow solvers are characterized by different mathematical formulations. The kinematic and dynamic fully nonlinear free surface boundary conditions are discussed, and in particular it is proven that the kinematic free surface boundary condition, written in semi-Lagrangian form, can be manipulated to derive an alternative non penetration boundary condition by all means identical to the one used on the surface of floating bodies or on the basin bottom. The simplified mathematical problem obtained is discretized over space and time via Boundary Element Method (BEM) and Implicit Backward Difference Formula (BDF) scheme, respectively. The results confirm that the solver implemented is able to solve steady potential flow problems just by eliminating null time derivatives in the unsteady formulation. Numerical results obtained confirm that the solver implemented is able to accurately reproduce results of classical steady flow solvers available in the literature.
Quantum algorithms for optimization problems are of general interest. Despite recent progress in classical lower bounds for nonconvex optimization under different settings and quantum lower bounds for convex optimization, quantum lower bounds for nonconvex optimization are still widely open. In this paper, we conduct a systematic study of quantum query lower bounds on finding $\epsilon$-approximate stationary points of nonconvex functions, and we consider the following two important settings: 1) having access to $p$-th order derivatives; or 2) having access to stochastic gradients. The classical query lower bounds is $\Omega\big(\epsilon^{-\frac{1+p}{p}}\big)$ regarding the first setting, and $\Omega(\epsilon^{-4})$ regarding the second setting (or $\Omega(\epsilon^{-3})$ if the stochastic gradient function is mean-squared smooth). In this paper, we extend all these classical lower bounds to the quantum setting. They match the classical algorithmic results respectively, demonstrating that there is no quantum speedup for finding $\epsilon$-stationary points of nonconvex functions with $p$-th order derivative inputs or stochastic gradient inputs, whether with or without the mean-squared smoothness assumption. Technically, our quantum lower bounds are obtained by showing that the sequential nature of classical hard instances in all these settings also applies to quantum queries, preventing any quantum speedup other than revealing information of the stationary points sequentially.
This paper presents a new distributed algorithm that leverages heavy-ball momentum and a consensus-based gradient method to find a Nash equilibrium (NE) in a class of non-cooperative convex games with unconstrained action sets. In this approach, each agent in the game has access to its own smooth local cost function and can exchange information with its neighbors over a communication network. The main novelty of our work is the incorporation of heavy-ball momentum in the context of non-cooperative games that operate on fully-decentralized, directed, and time-varying communication graphs, while also accommodating non-identical step-sizes and momentum parameters. Overcoming technical challenges arising from the dynamic and asymmetric nature of mixing matrices and the presence of an additional momentum term, we provide a rigorous proof of the geometric convergence to the NE. Moreover, we establish explicit bounds for the step-size values and momentum parameters based on the characteristics of the cost functions, mixing matrices, and graph connectivity structures. We perform numerical simulations on a Nash-Cournot game to demonstrate accelerated convergence of the proposed algorithm compared to that of the existing methods.
We propose an isogeometric solver for Poisson problems that combines i) low-rank tensor techniques to approximate the unknown solution and the system matrix, as a sum of a few terms having Kronecker product structure, ii) a Truncated Preconditioned Conjugate Gradient solver to keep the rank of the iterates low, and iii) a novel low-rank preconditioner, based on the Fast Diagonalization method where the eigenvector multiplication is approximated by the Fast Fourier Transform. Although the proposed strategy is written in arbitrary dimension, we focus on the three-dimensional case and adopt the Tucker format for low-rank tensor representation, which is well suited in low dimension. We show in numerical tests that this choice guarantees significant memory saving compared to the full tensor representation. We also extend and test the proposed strategy to linear elasticity problems.
We introduce a gradient-based approach for the problem of Bayesian optimal experimental design to learn causal models in a batch setting -- a critical component for causal discovery from finite data where interventions can be costly or risky. Existing methods rely on greedy approximations to construct a batch of experiments while using black-box methods to optimize over a single target-state pair to intervene with. In this work, we completely dispose of the black-box optimization techniques and greedy heuristics and instead propose a conceptually simple end-to-end gradient-based optimization procedure to acquire a set of optimal intervention target-state pairs. Such a procedure enables parameterization of the design space to efficiently optimize over a batch of multi-target-state interventions, a setting which has hitherto not been explored due to its complexity. We demonstrate that our proposed method outperforms baselines and existing acquisition strategies in both single-target and multi-target settings across a number of synthetic datasets.
Graph Neural Networks (GNNs) are often used for tasks involving the 3D geometry of a given graph, such as molecular dynamics simulation. Although the distance matrix of a geometric graph contains complete geometric information, it has been demonstrated that Message Passing Neural Networks (MPNNs) are insufficient for learning this geometry. In this work, we expand on the families of counterexamples that MPNNs are unable to distinguish from their distance matrices, by constructing families of novel and symmetric geometric graphs, to better understand the inherent limitations of MPNNs. We then propose $k$-DisGNNs, which can effectively exploit the rich geometry contained in the distance matrix. We demonstrate the high expressive power of $k$-DisGNNs from three perspectives: 1. They can learn high-order geometric information that cannot be captured by MPNNs. 2. They can unify some existing well-designed geometric models. 3. They are universal function approximators from geometric graphs to scalars (when $k\geq 2$) and vectors (when $k\geq 3$). Most importantly, we establish a connection between geometric deep learning (GDL) and traditional graph representation learning (GRL), showing that those highly expressive GNN models originally designed for GRL can also be applied to GDL with impressive performance, and that existing complex, equivariant models are not the only solution. Experiments verify our theory.
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