Numerical Formulation Research Articles

Coupling of multicomponent transport models in particle-resolved fluid-solid simulations

An accurate and self-consistent methodology for mass transport of multi-component mixtures in multi phase media is a necessity for a proper description of complex physical and chemical processes in reactors such as catalytic packed beds. In this regard, a novel methodology has been developed to describe and couple underlying transport phenomena in fluid and porous media as well as at the solid-fluid interface. The methodology is symmetric as it treats all components in a mixture equally. The Maxwell-Stefan equations are symmetrically formulated, discretized conservatively and coupled with a compressible flow solver for the fluid part. The Dusty Gas Model is applied inside porous media by developing a self-consistent and robust numerical formulation. A ghost-cell Immersed Boundary Method is used to capture the physics at the solid-fluid interface with the implementation of a novel symmetric non-singular mass flux formulation. Several test cases are established to demonstrate the accuracy and robustness of the newly developed symmetric methodology in this paper. These test cases can be used as benchmark for the future development of symmetric methodologies for multicomponent systems in multi phase media.

A numerical model for chemo-thermo-mechanical coupling at large strains with an application to thermoresponsive hydrogels

The aim of this work is the derivation and examination of a material model, accounting for large elastic deformations, coupled with species diffusion and thermal effects. This chemo-thermo-mechanical material model shows three key aspects regarding its numerical formulation. Firstly, a multiplicative split of the deformation gradient into a mechanical, a swelling and a thermal part. Secondly, temperature-scaled gradients for a numerical design comprising symmetric tangents and, thirdly, dissipation potentials for the modelling of dissipative effects. Additionally, the derived general material model is specialised to thermoresponsive hydrogels to study its predictive capabilities for a relevant example material class. An appropriate finite element formulation is established and its implementation discussed. Numerical examples are investigated, including phase transition and stability phenomena, to verify the ability of the derived chemo-thermo-mechanical material model to predict relevant physical effects properly. We compare our results to established models in the literature and discuss emerging deviations.

Thermomechanical performance analysis of MicroPCM-enhanced energy pile concrete: A multiscale numerical study and formulation design implications

Energy pile is a form of pile foundation that combines load-bearing and heat exchange, marking a pivotal avenue for harnessing geothermal energy. Recent strides in research have highlighted the incorporation of microencapsulated phase change materials (MicroPCM) into energy piles due to their potential to amplify heat transfer efficiency. The thermomechanical behavior of phase change energy piles, however, becomes intricate owing to the dual impact of thermal and mechanical loads. Thus, a holistic multiscale analysis remains indispensable to grasp the repercussions of varied MicroPCM concentrations on the thermomechanical attributes of phase change concrete and the dominant mechanisms driving their thermo-mechanical performance. In the study, we propose an innovative multiscale numerical model to investigate the thermomechanical performance of MicroPCM-enhanced energy pile concrete, taking into account the randomly distributed MicroPCM particles and aggregate. The sensitivity analysis pinpointing the intrinsic determinants of phase change concrete was conducted to discern the pivotal contributors to its thermomechanical response. Core insights revealed that while MicroPCM addition might compromise the mechanical strength of concrete owing to its inherently weaker properties, it unquestionably bolsters the thermal characteristics of phase change concrete. The thermal conductivity of phase change concrete stands in inverse relation to its porosity and direct correspondence with its saturation level. Correlation evaluations ascertained that the thermal prowess of phase change concrete is predominantly swayed by saturation, followed by porosity, and lastly, MicroPCM content. This inquiry endeavors to elevate the heat transfer efficacy of energy piles by finetuning the formulation design of phase change concrete materials, shedding light on underlying mechanisms, and laying a robust foundation for the forthcoming adoption of MicroPCM energy piles.

Numerical study of gas invasion law in fractured reservoirs

Gas invasion during the drilling process in fractured reservoirs poses challenges, affecting drilling efficiency and increasing costs. Therefore, it is crucial to effectively and accurately describe the flow characteristics of subsurface fluids. Addressing the issue of gas invasion in fractured reservoirs, this study considers the influence of matrix deformation and fracture aperture variation on fluid flow and establishes a mathematical model for coupled flow and solid deformation in fractured reservoirs. The numerical formulation of the mathematical model is derived using the finite element method. To better represent real reservoir conditions, discrete fractures are created using MATLAB, and numerical solutions are obtained using the commercial software COMSOL Multiphysics. The accuracy of the model is verified through a comparison between numerical and analytical solutions. This paper first explores the characteristics of fluid flow within a single fracture and rock deformation when encountering a fracture during drilling. It then compares the predictive capability of the coupled model with that of the uncoupled model in estimating gas invasion. Finally, the primary factors influencing gas invasion in fractured reservoirs are analyzed from the perspectives of rock matrix, fractures, and drilling operations.

Dent strain and stress analyses and implications concerning API RP 1183 - Part II: Examples of dent geometry and strain analyses during contact and re-rounding

API Recommended Practice (RP) 1183 considers three levels of assessment. Its Level 1 and Level 2 processes were considered viable for single peak dents with smooth profiles. The RP deals with more complex dents by way of a Level 3 approach that was reliant on finite element (FE) analysis.Part II of this four-part series of papers has identified the assumptions central to the practices of the RP, and evaluated them in regard to fully symmetric dents whose geometry is broadly aligned with those assumptions. Thereafter, it has examined the effects of asymmetry and skew angle benchmarked relative to the symmetric dents. It becomes apparent that even for symmetric dents significant errors emerge in the RP’s practices based on its reliance on dent profiles characterized along their axial and transverse axes cut through the apex, and the effects of the plastic deformation history developed in forming the dent. As for Part I, it was found that the practices of RP 1183 can (1) incorrectly categorize dents, and (2) grossly underestimate dent severity due to asymmetry and skew angles considered acceptable for Level 2 assessment. Error analyses and trending indicated conservative as well as nonconservative errors, with some more than 300%.As noted in Part I, Part III will consider cyclic loading of dents, and the viability of the dent stress and fatigue analyses that underlie the API RP 1183 Level 1 and Level 2 assessment practices, whereas Part IV considers the viability of the numerical formulations and modeling that underlie its practices.

Investigating hexagonal closed packed crystal lattice through QSPR modeling via linear regression analysis and Topsis

sThis research work covers three highly dominating reverse degree-based topological descriptors evaluated for the three-dimensional hexagonal close-packed (HCP) crystal structure lattice. HCP crystal lattice has a highly symmetrical and elegant crystal structure, and we have observed that results obtained from symmetrical structures draw symmetry in the numerical formulation obtained from topological descriptors. For this purpose, we will investigate whether the HCP lattice formed by an odd pair of unit cells is the best or an even pair of unit cells forming the lattice and how they behave in a dominating sense when investigated through the conclusions obtained from QSPR modeling. The QSPR model has been proposed to check the efficiency of each specified topological descriptor for five highly commendable physio-chemical properties: melting or boiling point, density, molar heat capacity, and enthalpy. Further, we have discovered the ideology to obtain the best possible structure when investigated and established on a statistically and mathematically strong proving and to what extent we can extract from QSPR to implement MCDM techniques, here considering TOPSIS that provide the best HCP structure lattice rankings. The obtained results may be helpful for researchers to better understand the structure and studying different physical/chemical properties of HCP crystal lattice.

Hybrid Mimetic Finite Difference and Streamline Methods for Numerical Simulation of Two-phase Flow in Fractured Reservoirs

In this work, we present a novel numerical formulation for efficiently modeling two-phase flow in fractured reservoirs. The novel formulation is developed by hybridizing mimetic finite difference and streamline (MFD-SL) methods, in which mimetic finite difference (MFD) is employed to discretize the pressure equation, and the streamline (SL) method is adopted to solve the saturation equation along 1D streamlines. The hybrid formulation is implemented on discrete fracture model (DFM) and is operated in an IMPES-like manner. A simple and practical streamline tracing method is developed on 2D triangular and 3D tetrahedral grids that are commonly implemented in the DFM framework. We benchmark the hybrid MFD-SL formulation with others using various cases with different complexity. Results show the proposed approach could achieve promising results yet with lower computation costs compared to other classical formulations. These achievements benefit from utilizing streamline tracing, which significantly alleviates the numerical diffusion error and improves the computation efficiency. This work, for the first time, incorporates hybrid MFD-SL formulation into DFM in case of triangular and tetrahedral grids and has great potential to be deployed on reservoir-scale simulation workflow.

Polymer composites with 3D graphene architectures as high-performance EMI shielding materials: a review

Polymer composites with a 3D graphene scaffold for EMI shielding are discussed. Numerical formulations to optimize the structural and geometrical parameters that are suitable to achieve an absorption-dominated shielding mechanism have been presented.

Morphological Investigation of Lamellae Patterns in Diblock Copolymers under Change of Thickness and Confinement in Polar Geometry

Mathematicians collaborate computationally with practical scientists to predict the novel morphologies soft materials, having potential applications in nanotechnology. This work presents the novel lamellae morphology that the confined diblock copolymer system displayed inside the polar geometry in order to achieve this, cell dynamics simulations (CDS) model is employed on a polar mesh to study the special impacts of confinement on the self-assembly behavior of diblock copolymer systems. Under the numerical formulation of the isotropic Laplacian with polar geometry employed in the CDS model, the set of 9-point stencil is considered to discretize the Laplacian operator on polar mesh system. FTCS finite difference mechanism is taken into consideration for discretization of Laplacian operator due to its computational efficiency and accuracy. Cell dynamics simulation is efficient simulation model for large-scale simulation as compared to the other models available in literature. In the simulation, evolutionary lamella morphology is presented with and without confinement for various sizes of the circular annular pores. A comparative review of lamella morphologies achieved by simulation in comparison of experimental study is presented, coupled with a thorough investigation of symmetric and asymmetric morphologies in captivity.

Numerical Method for Calculating Fourier Coefficients and Properties of Water Waves with Shear Current and Vorticity in Finite Depth

Many numerical methods have been developed since 1961, but unresolved issues remain. This study developed a numerical method to address these issues and determine the coefficients and properties of rotational waves with a shear current in a finite water depth. The number of unknown constants was reduced significantly by introducing a wavelength-independent coordinate system. The reference depth was calculated independently using the shooting method. Therefore, there was no need for partial derivatives with respect to the wavelength and the reference depth, which simplified the numerical formulation. This method had less than half of the unknown constants of the other method because Newton's method only determines the coefficients. The breaking limit was calculated for verification, and the result agreed with the Miche formula. The water particle velocities were calculated, and the results were consistent with the experimental data. Dispersion relations were calculated, and the results are consistent with other numerical findings. The convergence of this method was examined. Although the required series order was reduced significantly, the total error was smaller, with a faster convergence speed.

Investigation of combustion model via the local collocation technique based on moving Taylor polynomial (MTP) approximation/domain decomposition method with error analysis

In this paper, we develop a new meshless numerical procedure for simulating the combustion model. To that end, we employ a local meshless collocation method according to the moving Taylor polynomial (MTP) approximation. The space derivative is approximated by using the local approach and then the Crank–Nicolson algorithm is utilized to approximate the time derivative. The stability and convergence of the time-discrete formulation are discussed, analytically and numerically. The Broyden method is applied to solve this nonlinear system. Since the size of the physical domain is large, we employ the non-overlapping domain decomposition method (DDM) to obtain a faster numerical algorithm. The local meshless approaches are efficient numerical techniques to simulate models in the fluid flow. The obtained results show that the proposed numerical formulation has efficient results for solving this mathematical model.

An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods

We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation (slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.

A finite-element framework to explore the numerical solution of the coupled problem of heat conduction, water vapor diffusion, and settlement in dry snow (IvoriFEM v0.1.0)

Abstract. The poor treatment (or complete omission) of water vapor transport has been identified as a major limitation suffered by currently available snowpack models. As vapor and heat fluxes are closely intertwined, their mathematical representation amounts to a system of nonlinear and tightly coupled partial differential equations that are particularly challenging to solve numerically. The choice of the numerical scheme and the representation of couplings between processes are crucial to ensure an accurate and robust solution that guarantees mass and energy conservation while also allowing time steps in the order of 15 min. To explore the numerical treatments fulfilling these requirements, we have developed a highly modular finite-element program. The code is written in Python. Every step of the numerical formulation and solution is coded internally, except for the inversion of the linearized system of equations. We illustrate the capabilities of our approach to tackle the coupled problem of heat conduction, vapor diffusion, and settlement within a dry snowpack by running our model on several test cases proposed in recently published literature. We underline specific improvements regarding energy and mass conservation as well as time step requirements. In particular, we show that a fully coupled and fully implicit time-stepping approach enables accurate and stable solutions with little restriction on the time step.

A novel design of an array of pendulum-based electromagnetic broadband vibration energy harvester

This work investigates a novel design of an electromagnetic vibration energy harvester array based on pendulum structures under harmonic base excitation. The design includes combined magnetic-solenoid pendulum harvesters with different mechanical coupling configurations. A comparison is made with standalone magnetic and solenoid pendulum based harvester arrays under similar excitation conditions. Additionally, the positions of harvesters are shuffled and analyzed. The effects on harvested power, power density and frequency bandwidth are evaluated by considering dimensionless coupling ratios, damping ratios, mass ratios and length ratios as parameters. Further, the efficiency of the harvester against these parameters has also been reported. In addition, experimental validation of the numerical formulation is also conducted. The study shows that the proposed magnetic-solenoid harvester array can harvest sustained energy in a wider bandwidth compared to standalone magnetic and solenoid pendulum harvester arrays.

Effective thermo-electric-mechanical modeling of capacitively coupled plasma in low-pressure conditions: Modeling and application in dry etching

In semiconductor dry etching process, capacitively coupled plasma (CCP) is widely used in low-pressure environments. However, the kinetics of species particles in low-pressure conditions are different from the general environments. Traditional continuum models are popular in industry due to their simplicity, but they struggle to account for the kinetics of particles in low-pressure conditions. Although particle-in-cell (PIC) modeling is suitable for simulating the behavior of a single particle, it is difficult to perform such industrial simulations due to the high computational cost. To account for both kinetics and engineering applicability, this study presents an effective thermo-electric-mechanical modeling structure of CCP at low-pressure conditions. This modeling structure includes balance laws of continua derived from the global energy equilibrium containing thermal, electric and mechanical energies. The ion momentum, which is ignored in traditional plasma continuum models, is considered in the balance laws, and electron energy distributions are obtained by solving the Boltzmann equation to obtain the properties of electron transport. In addition, the model variables are calculated based on mixed kinetic theory accounting for the fraction of each species. The results of the proposed modeling are compared to those of a PIC model to validate its ability to describe the distribution of electrons with respect to pressure conditions. Based on the proposed numerical formulations, this study built a MATLAB-based program to guide the design of a screen baffle in dry etching equipment. This program captures the critical aspects of the practical application of screen baffle design in the context of the dry etching process. The results of this paper showed that the proposed model can be applied to engineering applications under low-pressure conditions.

The fracture failure modelling of three-dimensional structures composed of quasi-brittle materials subjected to different loading velocities rates by the dipole-based BEM approach

This study presents a numerical formulation for representing the fracture-failure of three dimensional structures composed of quasi-brittle materials. In addition to the nonlinear nature of quasi-brittle fracture, the formulation accounts the mechanical effects of different loading velocities rates. Therefore, these developments enable the modelling of viscous-cohesive fracture processes. The Boundary Element Method (BEM) describes the mechanical fields by the dipole-based approach, which has been extended to the three dimensional problems herein. This BEM approach is effective in crack growth modelling because only one crack surface requires discretisation. The classical cohesive crack approach models the material resistance at the energy dissipation zone surrounding the crack front. Besides, a time dependent function updates the cohesive crack approach and incorporates the viscous behaviour by modifying the tensile material strength and the material fracture energy as a function of the loading velocity rate. Three applications demonstrate the adequate performance of the proposed formulation, in which the results obtained have been compared against analytical and experimental responses available in the literature.

Further aliasing-reduced shifted and scaled Fresnel diffraction

Aliasing-reduced shifted and scaled Fresnel diffraction has been proposed to calculate the diffraction between parallel planes with different sampling intervals and an off-axis offset. This method suppresses the aliasing caused by scaling, that is, setting different sampling intervals, but does not suppress the aliasing caused by the shift. We herein propose further aliasing-reduced shifted and scaled Fresnel diffraction, which suppresses the aliasing caused by both scaling and shift. The proposed method is devised on the basis of the sampling theorem and the Fourier shift theorem. Numerical verification and theoretical formulations, which demonstrate the validity of the proposed method, are presented. Numerical verification showed that the increase in computation time was limited to a maximum of 4.7% and the signal-to-noise ratio of the diffraction image was improved by up to 37.4 dB compared to the conventional method.

Smart Elastic Material with a Moving Local Stiffness Zone for Reducing the Effects of Impact Loading

The elastic material properties which change momentarily and locally under the high deformation rate due to the movement of a wavefront are presented. The work contains mathematical formulation, semianalytical results, numerical formulations, and simulation results demonstrating the effectiveness of modifying the rheological properties of the elastic material upon shock load or contact with a rigid obstacle. While the semianalytical solutions can be obtained in a narrow time interval, numerical solutions allow us to track the process of wavefront reflections from edges. The effectiveness of reducing the physical quantities significant for impact in the presented examples reaches 30–70% of forces or accelerations, depending on the adopted criteria.

Ship grounding model tests in a water tank: An experimental study

Ship grounding experiments are important benchmarks used to validate numerical analysis, analytical and empirical formulation. They are key to the understanding of damage mechanism. A set of small-scale ship model grounding tests over a sharp rock are conducted in a water tank considering the influence of surrounding water. Two damage modes are observed in the grounding tests, Mode I for discontinuous fracture/tear and Mode II for continuous fracture/tear. The horizontal grounding resistance forces, damage extents of ship bottom plates, and ship motions are recorded and discussed in detail. Moreover, the energy dissipation process of ship model during grounding process is analyzed based on the test results. The influence of the initial velocity, the initial relative height between the upper surface of the horizontal ship bottom plate and the rock tip, and the rock eccentricity on the ship motion response and structural damage are studied.

Numerical and Dynamic Features for Heat Transfer Through a Longitudinal Rectangular Fin with Temperature-Dependent Thermal Conductivity

In the work reported herein, we adopt a Newton-Richtmeyer iterative scheme to arrive at solutions describing heat transfer in a longitudinal rectangular fin. Both the thermal conductivity and heat transfer coefficients are assumed to be temperature dependent and the governing partial differential equation highly nonlinear. In order to validate the numerical formulation, we consider some analytical results for this class of problems at steady state .Comparisons show convergence to exact solutions as demonstrated by local and global error indicators. In addition,various fin parameters which play a key role for efficient fin operation and design are investigated by carrying out a comprehensive dynamical qualitative study.