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This work presents a numerical formulation to model isotropic viscoelastic material behavior for membranes and thin shells. The surface and the shell theory are formulated within a curvilinear coordinate system, which allows the representation of general surfaces and deformations. The kinematics follow from Kirchhoff-Love theory and the discretization makes use of isogeometric shape functions. A multiplicative split of the surface deformation gradient is employed, such that an intermediate surface configuration is introduced. The surface metric and curvature of this intermediate configuration follow from the solution of nonlinear evolution laws - ordinary differential equations (ODEs) - that stem from a generalized viscoelastic solid model. The evolution laws are integrated numerically with the implicit Euler scheme and linearized within the Newton-Raphson scheme of the nonlinear finite element framework. The implementation of surface and bending viscosity is verified with the help of analytical solutions and shows ideal convergence behavior. The chosen numerical examples capture large deformations and typical viscoelasticity behavior, such as creep, relaxation, and strain rate dependence. It is shown that the proposed formulation can also be straightforwardly applied to model boundary viscoelasticity of 3D bodies.

This work proposes a model-reduction approach for the material point method on nonlinear manifolds. Our technique approximates the $\textit{kinematics}$ by approximating the deformation map using an implicit neural representation that restricts deformation trajectories to reside on a low-dimensional manifold. By explicitly approximating the deformation map, its spatiotemporal gradients -- in particular the deformation gradient and the velocity -- can be computed via analytical differentiation. In contrast to typical model-reduction techniques that construct a linear or nonlinear manifold to approximate the (finite number of) degrees of freedom characterizing a given spatial discretization, the use of an implicit neural representation enables the proposed method to approximate the $\textit{continuous}$ deformation map. This allows the kinematic approximation to remain agnostic to the discretization. Consequently, the technique supports dynamic discretizations -- including resolution changes -- during the course of the online reduced-order-model simulation. To generate $\textit{dynamics}$ for the generalized coordinates, we propose a family of projection techniques. At each time step, these techniques: (1) Calculate full-space kinematics at quadrature points, (2) Calculate the full-space dynamics for a subset of `sample' material points, and (3) Calculate the reduced-space dynamics by projecting the updated full-space position and velocity onto the low-dimensional manifold and tangent space, respectively. We achieve significant computational speedup via hyper-reduction that ensures all three steps execute on only a small subset of the problem's spatial domain. Large-scale numerical examples with millions of material points illustrate the method's ability to gain an order of magnitude computational-cost saving -- indeed $\textit{real-time simulations}$ -- with negligible errors.

This paper introduces a new simulation-based inference procedure to model and sample from multi-dimensional probability distributions given access to i.i.d. samples, circumventing the usual approaches of explicitly modeling the density function or designing Markov chain Monte Carlo. Motivated by the seminal work on distance and isomorphism between metric measure spaces, we propose a new notion called the Reversible Gromov-Monge (RGM) distance and study how RGM can be used to design new transform samplers to perform simulation-based inference. Our RGM sampler can also estimate optimal alignments between two heterogeneous metric measure spaces $(\mathcal{X}, \mu, c_{\mathcal{X}})$ and $(\mathcal{Y}, \nu, c_{\mathcal{Y}})$ from empirical data sets, with estimated maps that approximately push forward one measure $\mu$ to the other $\nu$, and vice versa. Analytic properties of the RGM distance are derived; statistical rate of convergence, representation, and optimization questions regarding the induced sampler are studied. Synthetic and real-world examples showcasing the effectiveness of the RGM sampler are also demonstrated.

This paper makes the first attempt to apply newly developed upwind GFDM for the meshless solution of two-phase porous flow equations. In the presented method, node cloud is used to flexibly discretize the computational domain, instead of complicated mesh generation. Combining with moving least square approximation and local Taylor expansion, spatial derivatives of oil-phase pressure at a node are approximated by generalized difference operators in the local influence domain of the node. By introducing the first-order upwind scheme of phase relative permeability, and combining the discrete boundary conditions, fully-implicit GFDM-based nonlinear discrete equations of the immiscible two-phase porous flow are obtained and solved by the nonlinear solver based on the Newton iteration method with the automatic differentiation, to avoid the additional computational cost and possible computational instability caused by sequentially coupled scheme. Two numerical examples are implemented to test the computational performances of the presented method. Detailed error analysis finds the two sources of the calculation error, roughly studies the convergence order thus find that the low-order error of GFDM makes the convergence order of GFDM lower than that of FDM when node spacing is small, and points out the significant effect of the symmetry or uniformity of the node collocation in the node influence domain on the accuracy of generalized difference operators, and the radius of the node influence domain should be small to achieve high calculation accuracy, which is a significant difference between the studied hyperbolic two-phase porous flow problem and the elliptic problems when GFDM is applied.

As the next-generation wireless networks thrive, full-duplex and relaying techniques are combined to improve the network performance. Random linear network coding (RLNC) is another popular technique to enhance the efficiency and reliability in wireless communications. In this paper, in order to explore the potential of RLNC in full-duplex relay networks, we investigate two fundamental perfect RLNC schemes and theoretically analyze their completion delay performance. The first scheme is a straightforward application of conventional perfect RLNC studied in wireless broadcast, so it involves no additional process at the relay. Its performance serves as an upper bound among all perfect RLNC schemes. The other scheme allows sufficiently large buffer and unconstrained linear coding at the relay. It attains the optimal performance and serves as a lower bound among all RLNC schemes. For both schemes, closed-form formulae to characterize the expected completion delay at a single receiver as well as for the whole system are derived. Numerical results are also demonstrated to justify the theoretical characterizations, and compare the two new schemes with the existing one.

This paper introduces a novel approach to compute the numerical fluxes at the cell boundaries for a cell-centered conservative numerical scheme. Explicit gradients used in deriving the reconstruction polynomials are replaced by high-order gradients computed by compact finite differences, referred to as implicit gradients in this paper. The new approach has superior dispersion and dissipation properties in comparison to the compact reconstruction approach. A problem-independent shock capturing approach via Boundary Variation Diminishing (BVD) algorithm is used to suppress oscillations for the simulation of flows with shocks and material interfaces. Several numerical test cases are carried out to verify the proposed method's capability using the implicit gradient method for compressible flows.

This paper proposes a numerical method based on the Adomian decomposition approach for the time discretization, applied to Euler equations. A recursive property is demonstrated that allows to formulate the method in an appropriate and efficient way. To obtain a fully numerical scheme, the space discretization is achieved using the classical DG techniques. The efficiency of the obtained numerical scheme is demonstrated through numerical tests by comparison to exact solution and the popular Runge-Kutta DG method results.

We introduce a finite volume scheme to solve isotropic 3-wave kinetic equations. We test our numerical solution against theoretical results concerning the long time behavior of the energy and observe that our solutions verify the energy cascade phenomenon. To our knowledge, this is the first numerical scheme that can capture the long time asymptotic behavior of solutions to isotropic 3-wave kinetic equations, where the energy cascade can be observed. Our numerical energy cascade rates are in good agreement with previously obtained theoretical results. The finite volume scheme given here relies on a new identity, allowing one to reduce the number of terms needed in the collision operators.

The minimum energy path (MEP) describes the mechanism of reaction, and the energy barrier along the path can be used to calculate the reaction rate in thermal systems. The nudged elastic band (NEB) method is one of the most commonly used schemes to compute MEPs numerically. It approximates an MEP by a discrete set of configuration images, where the discretization size determines both computational cost and accuracy of the simulations. In this paper, we consider a discrete MEP to be a stationary state of the NEB method and prove an optimal convergence rate of the discrete MEP with respect to the number of images. Numerical simulations for the transitions of some several proto-typical model systems are performed to support the theory.

We introduce a fast solver for the phase field crystal (PFC) and functionalized Cahn-Hilliard (FCH) equations with periodic boundary conditions on a rectangular domain that features the preconditioned Nesterov accelerated gradient descent (PAGD) method. We discretize these problems with a Fourier collocation method in space, and employ various second-order schemes in time. We observe a significant speedup with this solver when compared to the preconditioned gradient descent (PGD) method. With the PAGD solver, fully implicit, second-order-in-time schemes are not only feasible to solve the PFC and FCH equations, but also do so more efficiently than some semi-implicit schemes in some cases where accuracy issues are taken into account. Benchmark computations of five different schemes for the PFC and FCH equations are conducted and the results indicate that, for the FCH experiments, the fully implicit schemes (midpoint rule and BDF2 equipped with the PAGD as a nonlinear time marching solver) perform better than their IMEX versions in terms of computational cost needed to achieve a certain precision. For the PFC, the results are not as conclusive as in the FCH experiments, which, we believe, is due to the fact that the nonlinearity in the PFC is milder nature compared to the FCH equation. We also discuss some practical matters in applying the PAGD. We introduce an averaged Newton preconditioner and a sweeping-friction strategy as heuristic ways to choose good preconditioner parameters. The sweeping-friction strategy exhibits almost as good a performance as the case of the best manually tuned parameters.