In the present article we consider the problem of wave interaction with a partially immersed, but floating body. We assume that the motion of the body is prescribed. The general mathematical formulation for this problem is presented in the framework of a hierarchy of mathematical models. Namely, in this first part we formulate the problem at every hierarchical level. The special attention is paid to fully nonlinear and weakly dispersive models since they are most likely to be used in practice. For this model we have to consider separately the inner (under the body) and outer domains. Various approaches to the gluing of solutions at the boundary are discussed as well. We propose several strategies which ensure the global conservation or continuity of some important physical quantities.

Deep neural networks with rectified linear units (ReLU) are getting more and more popular due to their universal representation power and successful applications. Some theoretical progress regarding the approximation power of deep ReLU network for functions in Sobolev space and Korobov space have recently been made by [D. Yarotsky, Neural Network, 94:103-114, 2017] and [H. Montanelli and Q. Du, SIAM J Math. Data Sci., 1:78-92, 2019], etc. In this paper, we show that deep networks with rectified power units (RePU) can give better approximations for smooth functions than deep ReLU networks. Our analysis bases on classical polynomial approximation theory and some efficient algorithms proposed in this paper to convert polynomials into deep RePU networks of optimal size with no approximation error. Comparing to the results on ReLU networks, the sizes of RePU networks required to approximate functions in Sobolev space and Korobov space with an error tolerance ε, by our constructive proofs, are in general $O$($log\frac{1}{ε}$) times smaller than the sizes of corresponding ReLU networks constructed in most of the existing literature. Comparing to the classical results of Mhaskar [Mhaskar, Adv. Comput. Math. 1:61-80, 1993], our constructions use less number of activation functions and numerically more stable, they can be served as good initials of deep RePU networks and further trained to break the limit of linear approximation theory. The functions represented by RePU networks are smooth functions, so they naturally fit in the places where derivatives are involved in the loss function.

In this paper, we develop a conservative numerical method for the Cahn– Hilliard equation with generalized mobilities on curved surfaces in three-dimensional space. We use an unconditionally gradient stable nonlinear splitting numerical scheme and solve the resulting system of implicit discrete equations on a discrete narrow band domain by using a Jacobi-type iteration. For the domain boundary cells, we use the trilinear interpolation using the closest point method. The proposing numerical algorithm is computationally efficient because we can use the standard finite difference Laplacian scheme on three-dimensional Cartesian narrow band mesh instead of discrete Laplace–Beltrami operator on triangulated curved surfaces. In particular, we employ a mass conserving correction scheme, which enforces conservation of total mass. We perform numerical experiments on the various curved surfaces such as sphere, torus, bunny, cube, and cylinder to demonstrate the performance and effectiveness of the proposed method. We also present the dynamics of the CH equation with constant and space-dependent mobilities on the curved surfaces.

This study presents a new stochastic multiscale analysis approach to analyze the heat transfer performance of heterogeneous materials with random structures at different length scales. The heterogeneities of the materials are taken into account by periodic layouts of unit cells, consisting of randomly distributed inclusion dispersions and homogeneous matrix on the microscale and mesoscale. Based on the reiterated homogenization, a novel unified micro-meso-macro stochastic multiscale formulation is established and the scale gap is correlated by means of two-scale asymptotic expansions. Also, the stochastic multiscale formulae for computing the effective thermal property and temperature field are derived successively. Then, the stochastic prediction algorithm coupled with the finite element method is brought forward in details. The accuracy of the implemented stochastic multiscale analysis is verified by comparing the results against the experimental data for three scales heterogeneous materials with several different material combinations. The comparison demonstrates the usability of the proposed stochastic multiscale method for the determination of the thermal behaviors. This study offers a unified multiscale framework that enables heat transfer behavior analysis of heterogeneous materials with multiple random configurations.

RBF based grid-free scheme with PDE centres is experimented in this work for solving Convection-Diffusion Equations (CDE), a simplified model of the Navier-Stokes equations. For convection dominated problems, very few integration schemes can give converged solutions for the entire range of diffusivity wherein sharp layers are expected in the solutions and accurate computation of these layers is a big challenge for most of the numerical schemes. Radial Basis Function (RBF) based Local Hermitian Interpolation (LHI) with PDE centres is one such integration scheme which has some built in upwind effect and hence may be a good solver for the convection dominated problems. In the present work, to get convergent solutions consistently for small diffusion parameters, an explicit upwinding is also introduced in to the RBF based scheme with PDE centres, which was initially used to solve some time dependent problems in [10]. RBF based numerical schemes are one type of grid free numerical schemes based on the radial distances and hence very easy to use in high dimensional problems. In this work, the RBF scheme, with different upwind biasing, is used to a variety of steady benchmark problems with continuous and discontinuous boundary data and validated against the corresponding exact solutions. Comparisons of the solutions of the convective dominant benchmark problems show that the upwind biasing either in source centres or PDE centres gives convergent solutions consistently and is stable without any oscillations especially for problems with discontinuities in the boundary conditions. It is observed that the accuracy of the solutions is better than the solutions of other standard integration schemes particularly when the computations are carried out with fewer centers.

We study the two-component Camassa-Holm (2CH) equations as a model for the long time water wave propagation. Compared with the classical Saint-Venant system, it has the advantage of preserving the waves amplitude and shape for a long time. We present two different numerical methods—finite volume (FV) and hybrid finite-volume-particle (FVP) ones. In the FV setup, we rewrite the 2CH equations in a conservative form and numerically solve it by the central-upwind scheme, while in the FVP method, we apply the central-upwind scheme to the density equation only while solving the momentum and velocity equations by a deterministic particle method. Numerical examples are shown to verify the accuracy of both FV and FVP methods. The obtained results demonstrate that the FVP method outperforms the FV method and achieves a superior resolution thanks to a low-diffusive nature of a particle approximation.

A variety of numerical simulations based on continuum medium models have been carried out in unsteady flows to explore the physical characteristics qualitatively. However, there still lack quantitative valid scales for these models, in which scale the solutions by simulation can be treated as physically credible. In this work, Euler and Navier-Stokes models are implemented on the typical unsteady flow with discontinuities: Sod shock tube problem. Firstly, based on the Boltzmann model whose valid scale is molecule kinetic scale, we compare the values of different moments of Euler and Navier-Stokes to Boltzmann with the evolution of flow. The results provide the valid scales of Euler and Navier-Stokes models quantitatively. Secondly, following the real conditions and parameters in real air, the physical characteristics of waves’ generation and evolution in Sod shock tube are observed and analyzed numerically at the microscopic level for the first time. Our results present the reference scales for the choice of minimal simulation scale in which the continuum medium models are used, and for the criteria whether the flow characteristics obtained by simulations can be treated as real physical characteristics.

We design and analyze an efficient GPU-accelerated hybridizable discontinuous Galerkin method for linear elasticity. Performance analysis of the method is done using the state-of-the-art Time-Accuracy-Size (TAS) spectrum. TAS is a new performance measure which takes into account the accuracy of the solution. Standard performance measures, like floating point operations or run-time, are not completely appropriate for gauging the performance of approximations of continuum mechanics problems, as they neglect the solutions accuracy. A standard roofline model demonstrates that our method is utilizing computational resources efficiently, and as such, significant speed ups over a serial implementation are obtained. By combining traditional performance measures and the novel time-accuracy measures [7] into our performance model, we are able to draw more complete conclusions about which discretizations are best suited for an application. Several numerical experiments validate and verify our numerical scheme.

For solid-fluid interaction, one of the phasic density equations in diffuse interface models is degenerated to a "0 = 0" equation when the volume fraction of a certain phase takes the value of zero or unity, because the conservative variables in phasic density equations include volume fractions. The degeneracy can be avoided by adding an artificial quantity of another material into the pure phase. However, nonphysical waves are introduced by the artificial treatment. In this paper, an improved pressure-equilibrium diffuse interface model, which is able to treat zero/unity volume fractions, is presented for solid-fluid interaction. In the proposed model, the phasic density equations are replaced by the algebraic relation between phasic densities and inverse deformation gradient tensors. In consequence, the volume fractions and the phasic densities do not appear explicitly in the conservative equations any more. The degeneracy introduced by zero/unity volume fractions are prevented. A flux-splitting based finite difference algorithm suitable for this formulation is then presented. A series of one-dimensional and two-dimensional numerical tests demonstrate that the proposed model can present more accurate results near material interfaces.

In this work, the operator marching methods based on the modified Neumann-to-Dirichlet (NtD) map are developed to calculate the scattered fields from the periodic structures, especially diffraction gratings with metallic material. For the grating structures coated with the metallic material, the modified NtD map operator marching scheme is numerically stable for calculating the electromagnetic scattered fields. The modified NtD map operators are constructed by the integral equation (IE) method in each homogeneous medium of the layered medium structures, and avoid the blow up of the condition numbers from the challenging metallic material with the complex refractive index. For the calculations of the scattered fields from the diffraction grating structures, the modified NtD map operator marching method takes advantage of the periodic structure features, and avoids the calculations of the complicated quasi-periodic Green's function. Numerical results demonstrate that the proposed method achieves high accuracy and low computational workload for the scattered fields from both the dielectric and metallic diffraction grating structures.

We develop a novel numerical method for solving the nonlinear filtering problem of jump diffusion processes. The methodology is based on numerical approximation of backward stochastic differential equation systems driven by jump diffusion processes and we apply adaptive meshfree approximation to improve the efficiency of numerical algorithms. We then use the developed method to solve atom tracking problems in material science applications. Numerical experiments are carried out for both classic nonlinear filtering of jump diffusion processes and the application of nonlinear filtering problems in tracking atoms in material science problems.

Traditional two level upscaling techniques suffer from a high offline cost when the coarse grid size is much larger than the fine grid size one. Thus, multilevel methods are desirable for problems with complex heterogeneities and high contrast. In this paper, we propose a novel three-level upscaling method for flow problems in fractured porous media. Our method starts with a fine grid discretization for the system involving fractured porous media. In the next step, based on the fine grid model, we construct a nonlocal multi-continua upscaling (NLMC) method using an intermediate grid. The system resulting from NLMC gives solutions that have physical meaning. In order to enhance locality, the grid size of the intermediate grid needs to be relatively small, and this motivates using such an intermediate grid. However, the resulting NLMC upscaled system has a relatively large dimension. This motivates a further step of dimension reduction. In particular, we will apply the idea of the Generalized Multiscale Finite Element Method (GMsFEM) to the NLMC system to obtain a final reduced model. We present simulation results for a two-dimensional model problem with a large number of fractures using the proposed three-level method.