Solution of shallow-water equations using least-squares finite-element method

A least-squares finite-element method （LSFEM） for the non-conservative shallow-water equations is presented. The model is capable of handling complex topography, steady and unsteady flows, subcritical and supercritical flows, and flows with smooth and sharp gradient changes. Advantages of the model include： （1） sources terms, such as the bottom slope, surface stresses and bed frictions, can be treated easily without any special treatment; （2） upwind scheme is no needed; （3） a single approximating space can be used for all variables, and its choice of approximating space is not subject to the Ladyzhenskaya-Babuska-Brezzi （LBB） condition; and （4） the resulting system of equations is symmetric and positive-definite （SPD） which can be solved efficiently with the preconditioned conjugate gradient method. The model is verified with flow over a bump, tide induced flow, and dam-break. Computed results are compared with analytic solutions or other numerical results, and show the model is conservative and accurate. The model is then used to simulate flow past a circular cylinder. Important flow charac-teristics, such as variation of water surface around the cylinder and vortex shedding behind the cylinder are investigated. Computed results compare well with experiment data and other numerical results.

Least-squares finite-element method for shallow-water equations with source terms

Numerical solution of shallow-water equations （SWE） has been a challenging task because of its nonlinear hyperbolic nature, admitting discontinuous solution, and the need to satisfy the C-property. The presence of source terms in momentum equations, such as the bottom slope and friction of bed, compounds the difficulties further. In this paper, a least-squares finite-element method for the space discretization and θ-method for the time integration is developed for the 2D non-conservative SWE including the source terms. Advantages of the method include： the source terms can be approximated easily with interpolation functions, no upwind scheme is needed, as well as the resulting system equations is symmetric and positive-definite, therefore, can be solved efficiently with the conjugate gradient method. The method is applied to steady and unsteady flows, subcritical and transcritical flow over a bump, 1D and 2D circular dam-break, wave past a circular cylinder, as well as wave past a hump. Computed results show good C-property, conservation property and compare well with exact solutions and other numerical results for flows with weak and mild gradient changes, but lead to inaccurate predictions for flows with strong gradient changes and discontinuities.

Mesomechanics Finite-element Method for Determining the Shear Strength of Mudded Intercalation Materials

A new method regarding mesomechanics finite-element research is proposed to predict the peak shear strength of mudded intercalation materials on a mesoscopic scale. Based on geometric and mechanical parameters, along with the strain failure criteria obtained by sample＇s deformation characteristics, uniaxial compression tests on the sample were simulated through a finite-element model, which yielded values consistent with the data from the laboratory uniaxial compression tests, implying that the method is reasonable. Based on this model, a shear test was performed to calculate the peak shear strength of the mudded intercalation, consistent with values reported in the literature, thereby providing a new approach for investigating the mechanical properties of mudded intercalation materials.

TWO-GRID METHOD FOR CHARACTERISTICS FINITE-ELEMENT SOLUTION OF 2D NONLINEAR CONVECTION-DOMINATED DIFFUSION PROBLEM

For two-dimension nonlinear convection diffusion equation, a two-grid method of characteristics finite-element solution was constructed. In this method the nonlinear iterations is only to execute on the coarse grid and the fine-grid solution can be obtained in a single linear step. For the nonlinear convection-dominated diffusion equation, this method can not only stabilize the numerical oscillation but also accelerate the convergence and improve the computational efficiency. The error analysis demonstrates if the mesh sizes between coarse-grid and fine-grid satisfy the certain relationship, the two-grid solution and the characteristics finite-element solution have the same order of accuracy. The numerical is more efficient than that of characteristics example confirms that the two-grid method finite-element method.

Study on spline wavelet finite-element method in multi-scale analysis for foundation

A new finite element method （FEM） of B-spline wavelet on the interval （BSWI） is proposed. Through analyzing the scaling functions of BSWI in one dimension, the basic formula for 2D FEM of BSWI is deduced. The 2D FEM of 7 nodes and 10 nodes are constructed based on the basic formula. Using these proposed elements, the multiscale numerical model for foundation subjected to harmonic periodic load, the foundation model excited by external and internal dynamic load are studied. The results show the pro- posed finite elements have higher precision than the tradi- tional elements with 4 nodes. The proposed finite elements can describe the propagation of stress waves well whenever the foundation model excited by extemal or intemal dynamic load. The proposed finite elements can be also used to con- nect the multi-scale elements. And the proposed finite elements also have high precision to make multi-scale analysis for structure.

A Mixed Finite-Element Method on Polytopal Mesh

In this paper,we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem.We establish optimal order error estimates for velocity and super convergence for pressure.Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.

Pilot biomechanical design of biomaterials for artificial nucleus prosthesis using 3D finite-element modeling

Pilot biomechanical design of biomaterials for artificial nucleus prosthesiswas carried out based on the 3D finite-element method. Two 3D models of lumbar intervertebral discrespectively with a real human nucleus and with the nucleus removed were developed and validatedusing published experimental and clinical data. Then the models with a stainless steel nucleusprosthesis implanted and with polymer nucleus prostheses of various properties implanted were usedfor the 3D finite-element biomechanical analysis. All the above simulation and analysis were carriedout for the L4/L5 disc under a human worst--daily compression load of 2000 N. The results show thatthe polymer materials with Young's modulus of elasticity E = 0.1-100 MPa and Poisson's ratio v=0.35-0.5 are suitable to produce artificial nucleus prosthesis in view of biomechanicalconsideration.

The Calculation of Stress-Strain State of Anisotropic Composite Finite-Element Area with Different Boundary Conditions on the Surface

The numerical analytic research approach of stress-strain state of anisotropic composite finite element area with different boundary conditions on the surface, is represented below. The problem is solved by using a spatial model of the elasticity theory. Differential equation system in partial derivatives reduces to one-dimensional problem using spline collocation method in two coordinate directions. Boundary problem for the system of ordinary higher-order differential equation is solved by using the stable numerical technique of discrete orthogonalization.

Limit state analysis of rigid retaining structures against seismically induced passive failure in heterogeneous soils

Soils are not necessarily uniform and may present linearly varied or layered characteristics,for example the backfilled soils behind rigid retaining walls.In the presence of large lateral thrust imposed by arch bridge,passive soil failure is possible.A reliable prediction of passive earth pressure for the design of such wall is challenging in complicated soil strata,when adopting the conventional limit analysis method.In order to overcome the challenge for generating a kinematically admissible velocity field and a statically allowable stress field,finite element method is incorporated into limit analysis,forming finiteelement upper-bound(FEUB)and finite-element lower-bound(FELB)methods.Pseudo-static,original and modified pseudo-dynamic approaches are adopted to represent seismic acceleration inputs.After generating feasible velocity and stress fields within discretized elements based on specific criteria,FEUB and FELB formulations of seismic passive earth pressure(coefficient K_(P))can be derived from work rate balance equation and stress equilibrium.Resorting to an interior point algorithm,optimal upper and lower bound solutions are obtained.The proposed FEUB and FELB procedures are well validated by limit equilibrium as well as lower-bound and kinematic analyses.Parametric studies are carried out to investigate the effects of influential factors on seismic K_(P).Notably,true solution of K_(P) is well estimated based on less than 5%difference between FEUB and FELB solutions under such complex scenarios.

GENERALIZED WAVE EQUATION FINITE ELEMENT METHOD FOR SOLVING TWO-DIMENSIONAL TIDAL WAVES

The study of tidal circulation has a long history . The numerical simulation of tidal flow has been developed greatly with the development of computer techniques in the past two decades. The generalized wave equation finite-element method is a relatively new numerical model for studying shallow water flow . This method was used to simulate tidal waves of the Gulf of St. Lawrence in Canada . The very good agreement of the numerical results with the field data indicated that the model is an effective and promising numerical method for solving two-dimensional tidal wave problems .

Analysis of an Axial-flux Slotted Limited-angle Torque Motor with Quasi-Halbach Array for Torque Performance Improvement

Better torque performance and higher reliability have long been the focus of research for slotted limited-angle torque motors(LATMs),which are primarily used to position first-stage valves in electrohydraulic servosystems.This paper presents a high reliability axial-flux slotted LATM with quasi-Halbach array for torque performance improvement including constant torque range(CTR)and output torque.Firstly,the structure with two sets of windings and the operation principle of the proposed slotted LATM is analyzed.Secondly,a brief design procedure is presented,the structure selections of open slot and double-stator single-rotor(DSSR)interior rotor with surface mounted quasi-Halbach permanent magnet(PM)array are illustrated,and the geometric parameters are optimized to obtain the optimal design of the proposed slotted LATM.Then,3-D finite-element method(FEM)is employed to compare the proposed slotted LATM with the conventional surface mounted PM slotted LATM in terms of cogging torque,no-load back EMF,and output torque,and the results show that the proposed LATM with quasi-Halbach array has a 10%improvement in output torque and a 25%improvement in CTR.Meanwhile,the flux linkages and torque performance of the two sets of windings under various conditions verify good magnetic isolation.Finally,prototypes of two different rotor types are manufactured and a series of experiments are performed to validate the analysis.

Deep Learning Applied to Computational Mechanics:A Comprehensive Review,State of the Art,and the Classics

Three recent breakthroughs due to AI in arts and science serve as motivation:An award winning digital image,protein folding,fast matrix multiplication.Many recent developments in artificial neural networks,particularly deep learning(DL),applied and relevant to computational mechanics(solid,fluids,finite-element technology)are reviewed in detail.Both hybrid and pure machine learning(ML)methods are discussed.Hybrid methods combine traditional PDE discretizations with ML methods either(1)to help model complex nonlinear constitutive relations,(2)to nonlinearly reduce the model order for efficient simulation(turbulence),or(3)to accelerate the simulation by predicting certain components in the traditional integration methods.Here,methods(1)and(2)relied on Long-Short-Term Memory(LSTM)architecture,with method(3)relying on convolutional neural networks.Pure ML methods to solve(nonlinear)PDEs are represented by Physics-Informed Neural network(PINN)methods,which could be combined with attention mechanism to address discontinuous solutions.Both LSTM and attention architectures,together with modern and generalized classic optimizers to include stochasticity for DL networks,are extensively reviewed.Kernel machines,including Gaussian processes,are provided to sufficient depth for more advanced works such as shallow networks with infinite width.Not only addressing experts,readers are assumed familiar with computational mechanics,but not with DL,whose concepts and applications are built up from the basics,aiming at bringing first-time learners quickly to the forefront of research.History and limitations of AI are recounted and discussed,with particular attention at pointing out misstatements or misconceptions of the classics,even in well-known references.Positioning and pointing control of a large-deformable beam is given as an example.

Passive polarization rotator based on silica photonic crystal fiber for 1.31-μm and 1.55-μm bands via adjusting the fiber length

A new polarization rotator based on the silica photonic crystal fiber is proposed. The proposed polarization rotator photonic crystal fiber （PR-PCF） possesses a triangle jigsaw-shape core region. The full-vector finite-element method is used to analyze the phenomenon of polarization conversion between the quasi-TE and quasi-TM modes. Numerical simulations show that the wavelengths of 1.31 μm and 1.55 μm are converted with a nearly 100% polarization conversion ratio with their matched coupling length and has a relatively strong realistic fabrication tolerance - 100 nm on the y axis and 50 nm on the x axis. The full vectorial finite difference beam propagation method is used to confirm the performance of the proposed PR-PCF.

Fluid-solid Interaction Model for Hydraulic Reciprocating O-ring Seals

Elastohydrodynamic lubrication characteristics of hydraulic reciprocating seals have significant effects on sealing and tribology performances of hydraulic actuators, especially in high parameter hydraulic systems. Only elastic deformations of hydraulic reciprocating seals were discussed, and hydrodynamic effects were neglected in many studies. The physical process of the fluid-solid interaction effect did not be clearly presented in the existing fluid-solid interaction models for hydraulic reciprocating O-ring seals, and few of these models had been simultaneously validated through experiments. By exploring the physical process of the fluid-solid interaction effect of the hydraulic reciprocating O-ring seal, a numerical fluid-solid interaction model consisting of fluid lubrication, contact mechanics, asperity contact and elastic deformation analyses is constructed with an iterative procedure. With the SRV friction and wear tester, the experiments are performed to investigate the elastohydrodynamic lubrication characteristics of the O-ring seal. The regularity of the friction coefficient varying with the speed of reciprocating motion is obtained in the mixed lubrication condition. The experimental result is used to validate the fluid-solid interaction model. Based on the model, The elastohydrodynamic lubrication characteristics of the hydraulic reciprocating O-ring seal are presented respectively in the dry friction, mixed lubrication and full film lubrication conditions, including of the contact pressure, film thickness, friction coefficient, liquid film pressure and viscous shear stress in the sealing zone. The proposed numerical fluid-solid interaction model can be effectively used to analyze the operation characteristics of the hydraulic reciprocating O-ring seal, and can also be widely used to study other hydraulic reciprocating seals.

A finite-element simulation on deformational energy in structural deformed zone

Under the same conditions of external force, simulations on differences of deformational energy in structural zones, which have different deformational behavior, and that on the distribution of the differences have been carried out by means of finite-element method. Shear deformational energy U_w is higher than volume deformational energy U_T by about one order of magnitude, and deformationat energy U_B( = U_w + U_T) and U_w show a trend to be larger→largest→small in quantity in structural zones of different deformational properties, which correspond to compressive, shear, tensile zones, respectively, but U_T shows a trend to be large→small gradually. This has a considerable significance in the study of tectonic heat and mineral liquid migration in association with the research on tectonic additional hydrostatic pressure.

Numerical investigation of dual-porosity model with transient transfer function based on discrete-fracture model

Based on the characteristics of fractures in naturally fractured reservoir and a discrete-fracture model, a fracture network numerical well test model is developed. Bottom hole pressure response curves and the pressure field are obtained by solving the model equations with the finite-element method. By analyzing bottom hole pressure curves and the fluid flow in the pressure field, seven flow stages can be recognized on the curves. An upscaling method is developed to compare with the dual-porosity model （DPM）. The comparisons results show that the DPM overestimates the inter-porosity coefficient ）, and the storage factor w. The analysis results show that fracture conductivity plays a leading role in the fluid flow. Matrix permeability influences the beginning time of flow from the matrix to fractures. Fractures density is another important parameter controlling the flow. The fracture linear flow is hidden under the large fracture density. The pressure propagation is slower in the direction of larger fracture density.

Implicit scheme for integrating constitutive model of unsaturated soils with coupling hydraulic and mechanical behavior

A constitutive model of unsaturated soils with coupling capillary hystere- sis and skeleton deformation is developed and implemented in a fully coupled transient hydro-mechanical finite-element model （computer code U-DYSAC2）. The obtained re- sults are compared with experimental results, showing that the proposed constitutive model can simulate the main mechanical and hydraulic behavior of unsaturated soils in a unified framework. The non-lineaxity of the soil-water characteristic relation is treated in a similar way of elastoplasticity. Two constitutive relations axe integrated by an implicit return-mapping scheme similar to that developed for saturated soils. A consistent tan- gential modulus is derived to preserve the asymptotic rate of the quadratic convergence of Newton＇s iteration. Combined with the integration of the constitutive model, a complete finite-element formulation of coupling hydro-mechanical problems for unsaturated soils is presented. A number of practical problems with different given initial and boundary conditions are analyzed to illustrate the performance and capabilities of the finite-element model.

3D finite-element modeling of Earth induced electromagnetic field and its potential applications for geomagnetic satellites

The accumulated large amount of satellite magnetic data strengthens our capability of resolving the electrical conductivity of Earth’s mantle.To invert these satellite magnetic data,accurate and efficient forward modeling solvers are needed.In this study,a new finite-element based forward modeling solver is developed to accurately and efficiently compute the induced electromagnetic field for a realistic 3D Earth.Firstly,the nodal-based finite element method with linear shape function on tetrahedral grid is used to assemble the final system of linear equations for the magnetic vector potential and electric scalar potential.The FGMRES solver with algebraic multigrid(AMG)preconditioner is used to quickly solve the final system of linear equations.The weighted moving least-square method is employed to accurately recover the electromagnetic field from the numerical solutions of magnetic vector and electric scalar potentials.Furthermore,a local mesh refinement technique is employed to improve the accuracy of the estimated electromagnetic field.At the end,two synthetic models are used to verify the accuracy and efficiency of our newly developed forward modeling solver.A realistic 3D Earth model is used to simulate the induced magnetic field at 450 and 200 km altitudes which are the planned flying altitudes of Macao’s geomagnetic satellites.The simulation indicates that(1)the amplitude of the mantle-induced magnetic field can reach 10–30 nT at 450 km altitude,which is 10–30%of the primary magnetic field.The induced magnetic field at 200 km altitude has larger amplitudes.These mantleinduced magnetic fields can be measured by Macao geomagnetic satellites;(2)the amplitude of the ocean-induced magnetic field can reach 5–30 nT at satellite altitudes,which needs to be carefully considered in the interpretation of satellite magnetic data.We are confident that our newly developed forward modeling solver will become a key tool for interpreting satellite magnetic data.

Theoretical investigation of band-gap and mode characteristics of anti-resonance guiding photonic crystal fibres

With the full-vector plane-wave method （FVPWM） and the full-vector beam propagation method （FVBPM）, the dependences of the band-gap and mode characteristics on material index and cladding structure parameter in anti- resonance guiding photonic crystal fibres （ARGPCFs） are sufficiently analysed. An ARGPCF operating in the near- infrared wavelength is shown. The influences of the high index cylinders, glass interstitial apexes and silica structure on the characteristics of band-gaps and modes are deeply investigated. The equivalent planar waveguide theory is used for analysing such an ARGPCF filled by the isotropic materials, and the resonance and the anti-resonance characteristics r：~n h~ w~｜] r^r~dlrtpd

Micromechanical Analysis of Thermal Expansion Coefficients

Thermal expansion coefficients play an important role in the design and analysis of composite structures. A detailed analysis of thermo-mechanical distortion can be performed on microscopic level of a structure. However, for a design and analysis of large structures, the knowledge of effective material properties is essential. Thus, either a theoretical prediction or a numerical estimation of the effective properties is indispensable. In some simple cases, exact analytical solutions for the effective properties can be derived. Moreover, bounds on the effective values exist. However, in dealing with complex heterogeneous composites, numerical methods are becoming increasingly important and more widely used, because of the limiting applicability of the existing (semi-)analytical approaches. In this study, finite-element methods for the calculation of effective thermal expansion coefficients of composites with arbitrary geometrical inclusion configurations are discussed and applied to a heterogeneous lightning protection coating made from Dexmet&#174 copper foil 3CU7-100FA and HexPly&#174 epoxy resin M21. A short overview of some often used (semi-)analytical formulas for effective thermal expansion coefficients of heterogeneous composites is given in addition.