Free Boundary Conditions Research Articles

Physics-constrained neural networks for half-space seismic wave modeling

Forward modeling of seismic waves using physics-informed neural networks (PINNs) has attracted much attention. However, a notable challenge arises when modeling seismic wave propagation in large domains (i.e., a half-space), PINNs may encounter the issue of "soft constraint failure". To address this problem, we propose a novel framework called physics-constrained neural networks (PCNNs) specifically designed for modeling seismic wave propagation in a half-space. The method of images is incorporated to effectively implement the free stress boundary conditions of the Earth's surface, leading to the successful propagation of plane waves and cylindrical waves in a half-space. We analyze the training dynamics of neural networks when solving two-dimensional (2D) wave equations from the neural tangent kernel (NTK) perspective. An adaptive training algorithm is introduced to mitigate the unbalanced gradient flow dynamics of the different components of the loss function of PINNs/PCNNs. Furthermore, to tackle the complex behavior of seismic waves in layered media, a sequential training strategy is considered to enhance network scalability and solution accuracy. The results of numerical experiments demonstrate the accuracy and effectiveness of our approach.

The crust shell and the edge beams of third-gradient continua in current and referential description

A former publication has shown that some of the free boundary conditions of a third-gradient continuum can be interpreted as the equilibrium conditions of forces and moments of a crust shell and of edge beams. This was elaborated in the current (Eulerian) description. But it is obvious that a referential (Lagrangian) description of these physical phenomena must look very similar. This is, indeed, demonstrated in the paper at hand and disproves contrary claims in the literature. Moreover, the inductivist idea is omnipresent in the literature that a fictitious cut and a free surface show identical behaviour. But this is only correct with simple materials and not valid with higher gradient ones. We study a cube as a simple example where this wrong approach predicts the following absurd interaction between a subcube and the remainder of the cube: No actions at all appear on the fictitious cuts but there is one single force at the vertex of the subcube. Our findings do not depend on any specific constitutive assumption.

Determination of the effect of adhesive fillets and viscous damping on the dynamic response of aluminum honeycomb sandwich panels

The dynamic characteristics of an aluminum honeycomb sandwich panel are experimentally determined under free boundary conditions and the computational model is updated according to the experimental results. The study confirms that the adhesives used to bond the core and face sheets affect the dynamic response of the sandwich panel through the adhesive fillets formed between the face sheets and the core during the bonding process. The viscous damping of the structure is also determined by experimental modal analysis. It is observed that the results of the experimental and computational results agree well when the effects of adhesive fillets and viscous damping are considered. Furthermore, a dynamic response analysis under random vibration is conducted to reveal the differences between the results of the initial and the updated models.

GS-DeepNet: mastering tokamak plasma equilibria with deep neural networks and the Grad–Shafranov equation

The force-balanced state of magnetically confined plasmas heated up to 100 million degrees Celsius must be sustained long enough to achieve a burning-plasma state, such as in the case of ITER, a fusion reactor that promises a net energy gain. This force balance between the Lorentz force and the pressure gradient force, known as a plasma equilibrium, can be theoretically portrayed together with Maxwell’s equations as plasmas are collections of charged particles. Nevertheless, identifying the plasma equilibrium in real time is challenging owing to its free-boundary and ill-posed conditions, which conventionally involves iterative numerical approach with a certain degree of subjective human decisions such as including or excluding certain magnetic measurements to achieve numerical convergence on the solution as well as to avoid unphysical solutions. Here, we introduce GS-DeepNet, which learns plasma equilibria through solely unsupervised learning, without using traditional numerical algorithms. GS-DeepNet includes two neural networks and teaches itself. One neural network generates a possible candidate of an equilibrium following Maxwell’s equations and is taught by the other network satisfying the force balance under the equilibrium. Measurements constrain both networks. Our GS-DeepNet achieves reliable equilibria with uncertainties in contrast with existing methods, leading to possible better control of fusion-grade plasmas.

Exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries

Accelerated propagation is a new phenomenon associated with nonlocal diffusion problems. In this paper, we determine the exact rate of accelerated propagation in the Fisher-KPP equation with nonlocal diffusion and free boundaries, where the nonlocal diffusion operator is given by ∫RJ(x-y)u(t,y)dy-u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\displaystyle \\int _{\\mathbb {R}}J(x-y)u(t,y)dy-u(t,x)$$\\end{document}, and the kernel function J(x) behaves like a power function near infinity, namely lim|x|→∞J(x)|x|α=λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim _{|x|\\rightarrow \\infty } J(x)|x|^{\\alpha }=\\lambda >0$$\\end{document} for some α∈(1,2]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\in (1,2]$$\\end{document}. This is the precise range of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} where accelerated spreading can happen for such kernels. By constructing subtle upper and lower solutions, we prove that the location of the free boundaries x=h(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=h(t)$$\\end{document} and x=g(t)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x=g(t)$$\\end{document} goes to infinity at exactly the following rates: limt→∞h(t)tlnt=limt→∞-g(t)tlnt=μλ,whenα=2,limt→∞h(t)t1/(α-1)=limt→∞-g(t)t1/(α-1)=22-α2-αμλ,whenα∈(1,2).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle \\lim _{t\\rightarrow \\infty }\\frac{h(t)}{t\\ln t}=\\lim _{t\\rightarrow \\infty }\\frac{-g (t)}{t\\ln t}=\\mu \\lambda ,&{} \\hbox { when } \\alpha =2,\\\\ \\displaystyle \\lim _{t\\rightarrow \\infty }\\frac{h(t)}{ t^{1/(\\alpha -1)}}= \\lim _{t\\rightarrow \\infty }\\frac{-g (t)}{ t^{1/(\\alpha -1)}}=\\frac{2^{2-\\alpha }}{2-\\alpha }\\mu \\lambda , &{} \\hbox { when } \\alpha \\in (1,2). \\end{array}\\right. } \\end{aligned}$$\\end{document}Here μ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu >0$$\\end{document} is a given parameter in the free boundary condition. Accelerated propagation can also happen when lim|x|→∞J(x)|x|(ln|x|)β=λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lim _{|x|\\rightarrow \\infty }J(x)|x|(\\ln |x|)^\\beta =\\lambda >0$$\\end{document} for some β>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta >1$$\\end{document}. For this case, we prove that -g(t),h(t)=exp{[(2βμλβ-1)1/β+o(1)]t1/β}ast→∞.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -g (t), h(t)=\\exp \\Big \\{\\Big [\\Big (\\frac{2\\beta \\mu \\lambda }{\\beta -1}\\Big )^{1/\\beta }+o(1)\\Big ]t^{1/\\beta }\\Big \\} \\hbox { as } t\\rightarrow \\infty . \\end{aligned}$$\\end{document}These results considerably sharpen the corresponding ones in [20], and the techniques developed here open the door for obtaining similar precise results for other problems. A crucial technical point is that such precise conclusions on the propagation are achievable by finding the correct improvements on the form of the lower solutions used in [20], even though the precise long-time profile of the density function u(t, x) is still lacking.

Analytical solution for the bending problem of micropolar plates based on the symplectic approach

Abstract An analytical solution for the bending problem of micropolar plates is derived based on the symplectic approach. By applying Legendre's transformation, we obtain the Hamiltonian canonical equation for the bending problem of a micropolar plate. According to the method of separation of variables, the homogenous Hamiltonian canonical equation can be transformed into an eigenvalue problem of the Hamiltonian operator matrix. We derive the eigensolutions of the eigenvalue problem for the simply-supported, free and clamped boundary conditions at the two opposite sides. Based on the adjoint symplectic orthogonal relation of the eigensolutions, the solution of the bending problem of micropolar plate is expressed as a series expansion of eigensolutions. Numerical results confirm the validity of the present approach for the bending problem of micropolar plates under various boundary conditions and demonstrate the capability of the proposed approach to capture the size-dependent behavior of micropolar plates.

An improved acoustic/elastic interface approach for 2D staggered grid finite-difference modeling of Rayleigh waves in the presence of surface topography

Rayleigh waves are generated along the free-surface and their propagation can be strongly influenced by surface topography. Therefore, a critical aspect of Rayleigh-wave study is to understand the wave-propagation behavior in the presence of surface topography. The acoustic/elastic boundary approach was incorporated into a ‘stair-case’ mesh for Rayleigh-wave modeling in the presence of surface topography. The conventional acoustic/elastic boundary approach, however, fails to completely fulfill the free-surface boundary condition at some grid nodes, leading to unphysical simulation results. To solve this problem, we develop a stable acoustic/elastic boundary approach that fully satisfies the free-surface boundary condition by adjusting the free boundary conditions at problematic nodes in the mesh. The accuracy is demonstrated by modeling tests in two-dimensional isotropic models in the presence of surface topography. And compared to the improved vacuum formulation, the proposed method takes less time. Our approach optimizes the free-surface boundary condition and enables the high-precision numerical simulation of Rayleigh waves.

Simulation of flexural-gravity wave propagation for elastic plates in shallow water using an energy-stable finite difference method with weakly enforced boundary and interface conditions

We introduce an energy stable, high-order-accurate finite difference approximation of the dynamic, pure bending Kirchhoff plate equations for complex geometries and spatially variable properties. We utilize the summation-by-parts (SBP) framework to discretize the biharmonic operator with variable coefficients, with attention given to free and clamped boundary conditions and corner conditions. Energy conservation is established by combining SBP boundary closures with weak enforcement of the boundary and interface conditions using a penalty (simultaneous approximation term, SAT) technique. Then we couple the plate equations to the shallow water equations to study flexural-gravity wave propagation, and prove that the semi-discrete system of equations is self-adjoint. We demonstrate the stability and accuracy properties of the method on curvilinear multiblock grids using the method of manufactured solutions. The method, which we provide in an open-source code, is then used to model ocean wave interactions with the Thwaites Glacier and Pine Island Ice Shelf in the Amundsen Sea off the coast of West Antarctica.

MULTI-LINEAR-ELEMENT DEPTH-INTEGRATED MODELS FOR FLOWS WITH A FREE SURFACE

In this study, we derive a new set of depth-integrated models for solving unsteady flows that satisfy the Euler equations and nonlinear free surface boundary conditions in the sigma -coordinates. One of the obvious differences between the present approach and the ones directly solving the 3D Euler equations using Finite Element Method (FEM) is that in the present model the vertical velocity and the pressure field are eliminated by integrating the continuity equation and vertical momentum equation, respectively. Therefore, the present models only solve the horizontal velocity components and the free surface displacement in the two-dimensional horizontal (2DH) space. The new models are also more advantageous in using fewer elements in the vertical direction while achieving better performance. At this stage, the capability and application of the new models in dealing with free surface wave propagation problems are addressed.

The Effect of Nonlocal Scale Value and Phase Lags on Thermoelastic Waves in a Multilayered LEMV/CFRP Composite Cylinder

In this study, the effect of nonlocal scale value and two phase lags on the free vibration of generalized thermoelastic multilayered LEMV (Linear Elastic Material with Voids)/CFRP (Carbon Fiber Reinforced Polymer) composite cylinder is studied using nonlocal form of linear theory of elasticity. The governing equation of motion is established in longitudinal axis and variable separation model is used to transform the governing equations into a system of differential equations. To investigate vibration analysis from frequency equations, the stress free boundary conditions are adopted at the inner, outer and interface boundaries. The graphical representation of the numerically calculated results for frequency shift, natural frequency, and thermoelastic damping is presented. A special care has been taken to inspect the effect of nonlocal parameter on the aforementioned quantities. The results suggest that the nonlocal scale and the phase lag parameters alter the vibration characteristics of composite cylinders significantly.

Analysis of the vibrational characteristics of diamane nanosheet based on the Kirchhoff plate model and atomistic simulations

Single layer diamond—diamane, has been reported with excellent mechanical properties. In this work, molecular dynamics (MD) simulation and Kirchhoff plate model are utilized to investigate the vibrational characteristics of diamane sheets. The mechanical parameters of diamane sheets, including bending stiffness, Young’s modulus, Poisson’s ratio and coefficient of thermal expansion, are calibrated by using MD simulations. The natural frequencies and corresponding modal shapes of the diamane sheets predicted by the Kirchhoff plate model agree well with that obtained from the MD simulations. It is found that the edges exert marginal effect on the modal shapes when free boundary conditions are applied. Additionally, the Kirchhoff plate model considering the thermal expansion provides reasonable prediction for the natural frequencies of the diamane sheets with all boundary clamped under varying temperatures. This study offers valuable insights into the vibrational properties of diamane sheets, from both a simulation and theoretical standpoint. The findings would be beneficial for the design of nanoscale mechanical resonators utilizing these novel carbon materials.

Revealing the Sound Transmission Loss Capacities of Sandwich Metamaterials with Re-Entrant Negative Poisson's Ratio Configuration.

Due to the influence of mass law, traditional lightweight sandwich structures have struggled to surpass solid structures in sound insulation performance. To this end, we propose an acoustic metamaterial structure with a sandwich configuration based on the re-entrant negative Poisson's ratio (NPR) structure and systematically investigate its sound transmission loss (STL) performance under incident plane wave conditions. We used the acoustic impedance tube method to experimentally study the sound insulation performance of the re-entrant NPR sandwich structure under free boundary conditions, and then established an acoustic analysis simulation model based on COMSOL Multiphysics software, which verified that the results obtained by the experiment and the numerical simulation were in good agreement. The results show that the sandwich structure exhibits excellent sound transmission loss performance in the studied frequency range (250-4000 Hz), and the overall sound insulation performance exceeds the curve of the mass theorem, basically achieving more than 20 dB when the sandwich thickness is 2 cm. Finally, we conduct parametric studies to establish a correlation between the geometric design of NPR sandwich structures and their sound transmission loss performance. The research shows that the changes of the length of the ribs, the distance from the ribs to the center of the unit, and the length of the upper wall and the lower wall have a significant impact on the sound insulation performance of the re-entrant NPR sandwich structure, while the change of the wall thickness basically will not affect the sound insulation performance of the sandwich structure. This research can provide practical ideas for the engineering application of noise suppression designs of lightweight structures.

Pengaruh Pengerolan Alur Pelat Heksagonal Terhadap Frekuensi Pribadinya Dalam Kondisi Bebas-Bebas

Public interest in pick-up cars as a means of transportation for transporting goods has continued to increase lately because they are considered more flexible for transporting raw materials, workshop products and agricultural products and are capable of being operated on urban and rural roads that have different road roughness, as well as in very high traffic conditions. Plate panels that must be used in pick-up vehicles need to be designed as thin as possible but have high stiffness and natural frequency so that they are more efficient in fuel use and able to withstand vibrations that occur during driving. This study aims to determine the natural frequency that occurs in the plate panel profile for vehicle bodies resulting from the cold rolling process. The galvanized steel plates used are 0.6 and 0.8 mm thick. The natural frequency test method is used with free boundary conditions, where the conditioning plate is suspended by a rope. The natural frequency test tool used is the vibroport 80 at an existential pressure of 10 grams, with the hammer hitting the centre of the panel and the sensors being positioned at 0 and 90 degrees. Based on the test results that have been carried out, the panel specimen with a thickness of 0.8 mm obtained the highest natural frequency value of 922.5 Hz, while for a plate thickness of 0.6 mm, the highest value was only 816.25 Hz. However, the two groups of specimens with plate panel thicknesses of 0.6 and 0.8 mm because of rolling experienced an average increase in natural frequency, reaching 2.3 – 2.5 times when compared to the flat plates without the rolling process. The conclusion is that as the natural frequency value increases, the plate panel stiffness also increases.

Experimental research on damage identification of fully free beams based on modal shapes

Ship damage identification is a new field of ship research, which is of great significance to damage control, life and property safety and strategic decision-making, but there is no effective method for ship damage identification. The structural system changes caused by ship damage are usually represented by the change of system mass, stiffness and energy dissipation characteristics, and then the change of structural dynamic characteristic parameters, which provides a theoretical basis for ship damage identification. In this paper, the ship is simplified as a beam model with fully free boundary conditions. Based on the dynamic characteristic parameter of the modal shape, the damage identification of the ship under six damage conditions is carried out through numerical simulation and experimental verification by using COMAC and the derived indexes of curvature mode and modal flexibility. The results show that the first four orders of curvature mode difference mean index and the curvature index of modal flexibility change rate can accurately locate the damage of hull beam structure; The damage degree of hull beam can be identified by the change of curvature mode difference index, but there are some errors in this method; The curvature index of modal flexibility change rate is positively correlated with the damage degree, which can reflect the damage degree of the structure. The research results of this paper can provide new methods and ideas for hull beam damage identification.

Effect of free surface boundary condition on reflected waves in a half-space piezoelectric semiconductor medium

Effect of free surface boundary condition on reflected waves in a half-space piezoelectric semiconductor medium

Non-local Analyses of Tapered Beams

In the present article bending, vibration and buckling analyses of a tapered beam using Eringen non-local elasticity theory is being carried out. The associated governing differential equations are solved employing Rayleigh-Ritz method. Both Euler-Bernoulli and Timoshenko beam theories are considered in the analyses. Present results are in good agreement with those reported in literature. Non-local analyses for tapered beam with simply supported - simply supported (SS) , clamped - simply supported (CS) and clamped - free (CF) boundary conditions are conducted and discussed. It is observed that the maximum deflection increases with increase in non-local parameter value for SS and CS boundary conditions. Further, vibration frequency and critical buckling load decrease with increase in non-local parameter value for SS and CS boundary conditions. Non-local parameter effect on deflection, frequency and buckling load for CF supports is found to be opposite in nature to that of SS and CS supports. In case of thick beams non-local structural response is observed to be sensitive to length to thickness ratio.

3D Sensitivity Kernels With Full Attenuation Computed by a Combination of the Strong Stability Preserving Runge‐Kutta Method and the Scattering‐Integral Method

AbstractTo date developments of seismic attenuation models of the Earth have lagged those of velocity models. This is partly due to difficulties in isolating waveform perturbations caused by attenuation and velocity heterogeneities and partly due to different theories (e.g., ray theory, finite frequency theory) used in most previous and current studies to approximate sensitivity kernels and invert for the attenuation structure. We present in this paper a new method for computing the 3D waveform Fréchet kernels that account for full physical‐dispersion and dissipation attenuation. For solving the 3D isotropic anelastic wave equation described by the generalized Maxwell body model, we extend our previously proposed full waveform modeling method in Cartesian coordinates to that in spherical coordinates, which can provide stable numerical solutions even in the presence of strong attenuation. Then, we apply the scattering‐integral method for calculating 3D travel time and amplitude sensitivity kernels with respect to velocity and attenuation structures. We demonstrate the accuracy of our forward method and the effectiveness of the implementation of absorbing and free surface boundary conditions through numerical tests. Moreover, by choosing the Northwestern United States region as a realistic example, we verify the accuracy of the computed 3D sensitivity kernels through comparing the waveform measurements with predictions from the kernels. Finally, we discuss the importance of calculating full anelastic sensitivity kernels including both effects of physical dispersion and dissipation, where we specially explore the effect of scattering due to random velocity and attenuation heterogeneities on waveform measurements.

Brinkman–Bénard Convection with Rough Boundaries and Third-Type Thermal Boundary Conditions

The Brinkman–Bénard convection problem is chosen for investigation, along with very general boundary conditions. Using the Maclaurin series, in this paper, we show that it is possible to perform a relatively exact linear stability analysis, as well as a weakly nonlinear stability analysis, as normally performed in the case of a classical free isothermal/free isothermal boundary combination. Starting from a classical linear stability analysis, we ultimately study the chaos in such systems, all conducted with great accuracy. The principle of exchange of stabilities is proven, and the critical Rayleigh number, Rac, and the wave number, ac, are obtained in closed form. An asymptotic analysis is performed, to obtain Rac for the case of adiabatic boundaries, for which ac≃0. A seemingly minimal representation yields a generalized Lorenz model for the general boundary condition used. The symmetry in the three Lorenz equations, their dissipative nature, energy-conserving nature, and bounded solution are observed for the considered general boundary condition. Thus, one may infer that, to obtain the results of various related problems, they can be handled in an integrated manner, and results can be obtained with great accuracy. The effect of increasing the values of the Biot numbers and/or slip Darcy numbers is to increase, not only the value of the critical Rayleigh number, but also the critical wave number. Extreme values of zero and infinity, when assigned to the Biot number, yield the results of an adiabatic and an isothermal boundary, respectively. Likewise, these extreme values assigned to the slip Darcy number yield the effects of free and rigid boundary conditions, respectively. Intermediate values of the Biot and slip Darcy numbers bridge the gap between the extreme cases. The effects of the Biot and slip Darcy numbers on the Hopf–Rayleigh number are, however, opposite to each other. In view of the known analogy between Bénard convection and Taylor–Couette flow in the linear regime, it is imperative that the results of the latter problem, viz., Brinkman–Taylor–Couette flow, become as well known.

PolyStokes: A Polynomial Model Reduction Method for Viscous Fluid Simulation

Standard liquid simulators apply operator splitting to independently solve for pressure and viscous stresses, a decoupling that induces incorrect free surface boundary conditions. Such methods are unable to simulate fluid phenomena reliant on the balance of pressure and viscous stresses, such as the liquid rope coil instability exhibited by honey. By contrast, unsteady Stokes solvers retain coupling between pressure and viscosity, thus resolving these phenomena, albeit using a much larger and thus more computationally expensive linear system compared to the decoupled approach. To accelerate solving the unsteady Stokes problem, we propose a reduced fluid model wherein interior regions are represented with incompressible polynomial vector fields. Sets of standard grid cells are consolidated into super-cells, each of which are modelled using a quadratic field of 26 degrees of freedom. We demonstrate that the reduced field must necessarily be at least quadratic, with the affine model being unable to correctly capture viscous forces. We reproduce the liquid rope coiling instability, as well as other simulated examples, to show that our reduced model is able to reproduce the same fluid phenomena at a smaller computational cost. Futhermore, we performed a crowdsourced user survey to verify that our method produces imperceptible differences compared to the full unsteady Stokes method.

Numerical modeling on global-scale mantle water cycle and its impact on the sea-level change

Using numerical modeling of the global-scale mantle water cycle, I have clarified the mechanisms governing the deep Earth water content variations and evaluated their effects on sea-level changes. The following points are found: 1. The temporal variations of water content in the deep Earth are mainly determined by a balance between the ingassing flux from the ocean and the dehydration flux in the mantle wedge. 2. The primary control of ingassing flux comes from the water content of oceanic crust, rather than the style of plate subduction promoted by the free surface boundary condition. 3. The water content of oceanic crust ranges from 0.1 to 1 wt. percent to match the ingassing flux with the inferred one from the geological analysis.Based on the best-fit scenario for the ingassing flux from the ocean to the deep interior, I obtain an average rate of sea-level change of −1 to −1.5 m/Myrs over 1 billion year, which is nearly consistent with the long-term trend of sea-level change estimated from geological and paleontological data (∼−1 m/Myr). Note, however, that the detailed profile of sea-level change in the numerical modeling is difficult to explain the observation one. This underlines the importance of dehydration process in the mantle wedge in understanding the sea-level change resulting from the water budget in the Earth's deep interior over the Earth's history.