Enriched reproducing kernel particle method for fractional advection–diffusion equation

The reproducing kernel particle method （RKPM） has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advectiondiffusion equation （FADE）, which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM, leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.

Artificial boundary conditions for out-of-plane motion in penta-graphene

In this paper, artificial boundary conditions are designed for out-of-plane waves in penta-graphene, a newly proposed allotrope of carbon. By matching the dispersion relation for acoustic branch phonons in the long wave limit, we determine parameters in proposed linear constraints among displacements and velocities at the boundary and nearby atoms. Reflection analysis for normal incidences and a numerical test demonstrate the effectiveness of the artificial boundary conditions. These conditions may be used for studying mechanical behaviours of the novel complex lattice of penta-graphene. ? 2017 The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg

Stable heat jet approach for temperature control of Fermi-Pasta-Ulam beta chain

In this paper,we propose a stable heat jet approach for accurate temperature control of the nonlinear Fermi-Pasta-Ulam beta(FPU-β)chain.First,we design a stable nonlinear boundary condition,with co-efficients determined by a machine learning technique.Its stability can be proved rigorously.Based on this stable boundary condition,we derive a two-way boundary condition complying with phonon heat source,and construct stable heat jet approach.Numerical tests illustrate the stability of the boundary condition and the effectiveness in eliminating boundary reflections.Furthermore,we extend the bound-ary condition formulation with more atoms,and train the coefficients to eliminate extreme short waves by machine learning technique.Under this extended boundary condition,the heat jet approach is effec-tive for high temperature,and may be adopted for multiscale computation of atomic motion at finite temperature.

Mechanistic Machine Learning:Theory,Methods,and Applications

Recent advances in machine learning are currently influencing the way we gather data,recognize patterns,and build predictive models across a wide range of scientific disciplines.Noticeable successes include solutions in image and voice recognition that have already become part of our everyday lives,mainly enabled by algorithmic developments,hardware advances,and,of course,the availability of massive data-sets.Many of such predictive tasks are currently being tackled using over-parameterized,black-box discriminative models such as deep neural networks,in which theoretical rigor,interpretability and adherence to first physical principles are often sacrificed in favor of flexibility in representation and scalability in computation.

Displacement reconstruction and strain refinement of clustering-based homogenization

Recently proposed clustering-based methods provide an efficient way for homogenizing heterogeneous materials,yet without concerning the detailed distribution of the mechanical responses.With coarse fields of the clustering-based methods as an initial guess,we develop an iteration strategy to fastly and accurately resolve the displacement,strain and stress based on the Lippmann-Schwinger equation,thereby benefiting the local mechanical analysis such as the detection of the stress concentration.From a simple elastic case,we explore the convergence of the method and give an instruction for the selection of the reference material.Numerical tests show the efficiency and fast convergence of the reconstruction method in both elastic and hyper-elastic materials.

Clustering solver for displacement-based numerical homogenization

Based on strain-clustering via k-means,we decompose computational domain into clusters of possibly disjoint cells.Assuming cells in each cluster take the same strain,we reconstruct displacement field.We further propose a new way to condensate the governing equations of displacement-based finite element method to reduce the complexity while maintain the accuracy.Numerical examples are presented to illustrate the efficiency of the clustering solver.Numerical convergence studies are performed for the examples.Avoiding complexities which is common in existing clustering analysis methods,the proposed clustering solver is easy to implement,particularly for numerical homogenization using commercial softwares.

Why neural networks apply to scientific computing?

In recent years,neural networks have become an increasingly powerful tool in scientific computing.The universal approximation theorem asserts that a neural network may be constructed to approximate any given continuous function at desired accuracy.The backpropagation algorithm further allows efficient optimization of the parameters in training a neural network.Powered by GPU’s,effective computations for scientific and engineering problems are thereby enabled.In addition,we show that finite element shape functions may also be approximated by neural networks.

Matching Boundary Conditions for Scalar Waves in Body-Centered-Cubic Lattices

Matching boundary conditions(MBC’s)are proposed to treat scalar waves in the body-centered-cubic lattices.By matching the dispersion relation,we construct MBC’s for normal incidence and incidence with an angle a.Multiplication of MBC operators then leads to multi-directional absorbing boundary conditions.The effectiveness are illustrated by the reflection coefficient analysis and wave packet tests.In particular,the designed M1M1 treats the scalar waves in a satisfactory manner.

A Two-Way Interfacial Condition for Lattice Simulations

In this paper,we formulate a two-way interfacial condition for simulating lattice dynamics in one space dimension.With a time history treatment,the incoming waves are incorporated into the motion of the boundary atoms accurately.This condition reduces to the absorbing boundary condition when there is no incoming wave.Numerical tests validate the effectiveness of the proposed condition in treating simultaneously incoming waves and outgoing waves.

Stationary and Transient Simulations for a One-Dimensional Resonant Tunneling Diode

We investigate the validity of stationary simulations for semiconductor quantum charge transport in a one-dimensional resonant tunneling diode via fluid type models.Careful numerical investigations to a quantum hydrodynamic model reveal that the transient simulations do not always converge to the steady states.In particular,growing oscillations are observed at relatively large applied voltage.A dynamical bifurcation is responsible for the stability interchange of the steady state.Transient and stationary computations are also performed for a unipolar quantum drift-diffusion model.

Unstable Surface Modes in Finite Chain Computations:Deficiency of Reflection Coefficient Approach

In this paper,we investigate the stability for a finite harmonic lattice under a certain class of boundary conditions.A rigorous eigenvalue study clarifies that the invalidity of Fourier modes as the basis results in the deficiency of standard reflection coefficient approach for stability analysis.In a certain parameter range,unstable surface modes exist in the form of exponential decay in space,and exponential growth in time.An approximate eigen-polynomial is proposed to ease the stability analysis.Moreover,the eigenvalues with small positive real part quantitatively explain the long time instability in wave propagation computations.Numerical results verify the analysis.