Internal Boundary Conditions Research Articles

Effect of crack location on vibration analysis of partially cracked isotropic and FGM micro-plate with non-uniform thickness: An analytical approach

An analytical model is presented for nonlinear vibration analysis of variable thickness, thin isotropic and functionally graded rectangular micro-plate containing a partial crack located within the centre line of the plate. Linear and parabolic thickness variation is considered in one or both the in-plane directions of the plate. The thickness variation is such that the volume of the plate is equal to that of the uniform thickness plate. The continuous line crack is parallel to one of the edges of the plate. The equations of motion are derived using the equilibrium principle based on Classical Plate Theory while the size effect is incorporated using the modified couple stress theory. The partial crack is represented by bending moment and in-plane force according to the simplified line spring model. The effect of in-plane forces is considered by employing the Berger's formulation. The derived governing equation is converted into a cubic nonlinear Duffing equation by employing Galerkin's method. The effect of nonlinearity is established by deriving the frequency response equation for the cracked variable thickness plate using the method of multiple scales. The nonlinear frequency response curves show the phenomenon of bending hardening or softening. The influence of crack length, crack location, internal material length scale parameter, boundary conditions, gradient index, unidirectional and bidirectional taper constant on the fundamental frequency of square plate is demonstrated. Similar to uniform thickness plate, it is found that the presence of crack affects the vibration characteristics of variable thickness plate. When compared to uniform thickness plate, the effect of crack can be reduced by varying the thickness of the plate. Thus, it is concluded that in a given volume, it is better to employ variable thickness plate as far as the vibration characteristics are considered.

Towards a comprehensive model for characterising and assessing thermoelectric modules by impedance spectroscopy

Thermoelectric devices have potential energy conversion applications ranging from space exploration through to mass-market products. Standardised, accurate and repeatable high-throughput measurement of their properties is a key enabling technology. Impedance spectroscopy has shown promise as a tool to parametrically characterise thermoelectric modules with one simple measurement. However, previously published models which attempt to characterise fundamental properties of a thermoelectric module have been found to rely on heavily simplified assumptions, leaving its validity in question. In this paper a new comprehensive impedance model is mathematically developed. The new model integrates all relevant transport phenomena: thermal convection, radiation, and spreading-constriction at junction interfaces. Additionally, non-adiabatic internal surface boundary conditions are introduced for the first time. These phenomena were found to significantly alter the low and high frequency response of Nyquist spectra, showing their necessity for accurate characterisation. To validate the model, impedance spectra of a commercial thermoelectric module was experimentally measured and parametrically fitted. Technique precision is investigated using a Monte-Carlo residual resampling approach. A complete characterisation of all key thermoelectric properties as a function of temperature is validated with material property data provided by the module manufacturer. Additionally, by firstly characterising the module in vacuum, the ability to characterise a heat transfer coefficient for free and forced convection is demonstrated. The model developed in this study is therefore a critical enabler to potentially allow impedance spectroscopy to characterise and monitor manufacturing and operational defects in practical thermoelectric modules across multiple sectors, as well as promote new sensor technologies.

Stability analysis and observer design for one-sided Lipschitz descriptor systems with time-varying delay

This paper investigates the problem of stability analysis and observer design for nonlinear descriptor systems with time-varying delay. In the systems, the nonlinear function satisfies the one-sided Lipschitz condition and the quadratic internal boundary condition. The disturbance is considered in both the state and the output equation. Using one-sided Lipschitz condition, the quadratic internal boundary condition, and the generalized Lyapunov method, we establish the non-strict bilinear matrix inequality (BMI)-based condition. We change the condition into strict bilinear matrix inequality (BMI) condition. Furthermore, we give the linear matrix inequality-based condition to ensure the gradual convergence of state estimation error and to accomplish robustness against L2 norm bounded disturbances by utilizing change of variables for straight forward computation of the observer gain matrix. Finally, a numerical example is given to verify the effectiveness of the observer design scheme.

Uniform stabilization of wave equation with localized internal damping and acoustic boundary condition with viscoelastic damping

In this paper, we deal with the uniform stabilization to the mixed problem for a nonlinear wave equation and acoustic boundary conditions on a non-locally reacting boundary. The main purpose is to study the stability when the internal damping acts only over a subset $$\omega $$ of the domain $$\Omega $$ and the boundary damping is of the viscoelastic type.

Semi-analytical solution of multilayer diffusion problems with time-varying boundary conditions and general interface conditions

We develop a new semi-analytical method for solving multilayer diffusion problems with time-varying external boundary conditions and general internal boundary conditions at the interfaces between adjacent layers. The convergence rate of the semi-analytical method, relative to the number of eigenvalues, is investigated and the effect of varying the interface conditions on the solution behaviour is explored. Numerical experiments demonstrate that solutions can be computed using the new semi-analytical method that are more accurate and more efficient than the unified transform method of Sheils [Appl. Math. Model., 46:450–464, 2017]. Furthermore, unlike classical analytical solutions and the unified transform method, only the new semi-analytical method is able to correctly treat problems with both time-varying external boundary conditions and a large number of layers. The paper is concluded by replicating solutions to several important industrial, environmental and biological applications previously reported in the literature, demonstrating the wide applicability of the work.

Characteristic time scales for diffusion processes through layers and across interfaces.

This paper presents a simple tool for characterizing the time scale for continuum diffusion processes through layered heterogeneous media. This mathematical problem is motivated by several practical applications such as heat transport in composite materials, flow in layered aquifers, and drug diffusion through the layers of the skin. In such processes, the physical properties of the medium vary across layers and internal boundary conditions apply at the interfaces between adjacent layers. To characterize the time scale, we use the concept of mean action time, which provides the mean time scale at each position in the medium by utilizing the fact that the transition of the transient solution of the underlying partial differential equation model, from initial state to steady state, can be represented as a cumulative distribution function of time. Using this concept, we define the characteristic time scale for a multilayer diffusion process as the maximum value of the mean action time across the layered medium. For given initial conditions and internal and external boundary conditions, this approach leads to simple algebraic expressions for characterizing the time scale that depend on the physical and geometrical properties of the medium, such as the diffusivities and lengths of the layers. Numerical examples demonstrate that these expressions provide useful insight into explaining how the parameters in the model affect the time it takes for a multilayer diffusion process to reach steady state.

Size effect on cracked functional composite micro-plates by an XIGA-based effective approach

Failure of structures and their components is one of major problems in engineering. Studies on mechanical behavior of functionally graded (FG) microplates with defects or cracks by effective numerical methods are rarely reported in literature. In this paper, an effective numerical model is derived based on extended isogeometric analysis (XIGA) for assessment of vibration and buckling of FG microplates with cracks. Based on the modified couple stress theory, the non-classical theory of Reissner–Mindlin plate is extended to capture microstructure, and thus, the size effect. In such theory, possessing C1-continuity is straightforward with the high-order continuity of non-uniform rational B-spline. Due to the use of enrichments in XIGA, crack geometry is independent of the computational mesh. Numerical examples are performed to illustrate the effects of microplate aspect ratio, crack length, internal material length scale parameter, material distribution, and boundary condition on the mechanical responses of cracked FG microplates. The obtained results are compared with reference solutions and that shows that the frequency and buckling loads increases with decreasing the size of FG microplates and crack length. The convergence of the present method is also studied.

An explicit algorithm for imbedding solid boundaries in Cartesian grids for the reactive Euler equations

We present a local and point-wise scheme for imposing reflective boundary conditions to stationary internal boundaries for solving the reactive Euler equations on Cartesian grids. The scheme is presented in two and three dimensions and can run efficiently on parallel machines while still maintaining the same advantages over other methods for enforcing internal boundary conditions. Level sets are used to represent internal solid regions along with a new local node sorting algorithm that decouples internal boundary nodes by establishing their connectivity to other internal boundary nodes. This approach allows us to enforce boundary conditions via a direct procedure, removing the need to solve a coupled system of equations numerically. We examine the accuracy and fidelity of our internal boundary algorithm by simulating flows past various solid boundaries in two and three dimensions, showing good agreement between our numerical results and experimental data.

A finite-volume discretization for deformation of fractured media

Simulating the deformation of fractured media requires the coupling of different models for the deformation of fractures and the formation surrounding them. We consider a cell-centered finite-volume approach, termed the multi-point stress approximation (MPSA) method, which is developed in order to discretize coupled flow and mechanical deformation in the subsurface. Within the MPSA framework, we consider fractures as co-dimension one inclusions in the domain, with the fracture surfaces represented as line pairs in 2D (face pairs in 3D) that displace relative to each other. Fracture deformation is coupled to that of the surrounding domain through internal boundary conditions. This approach is natural within the finite-volume framework, where tractions are defined on surfaces of the grid. The MPSA method is capable of modeling deformation, considering open and closed fractures with complex and nonlinear relationships governing the displacements and tractions at the fracture surfaces. We validate our proposed approach using both problems, for which analytical solutions are available, and more complex benchmark problems, including comparison with a finite-element discretization.

Effect of bottom surface optical boundary conditions on nanofluid-based DASC: Parametric study and optimization

In this study, low-flux direct absorption solar collectors (DASCs) with nanofluid volume absorbers were modeled, analyzed, and optimized. The Rayleigh scattering approximation with size-dependent effects was used in order to determine nanofluid optical properties. Upon validating the mathematical model resulting from numerically solving and coupling the energy conservation equation with the radiative transfer equation, effects of internal bottom-surface optical boundary condition and base-fluid type on nanofluid temperature homogeneity and collector first- and second-law efficiencies were studied for different particle loadings, film thicknesses, and nanoparticle materials. Non-linear multi-variable constrained single- and multi-objective global optimization studies were conducted to find the optimal design vectors with respect to first- and/or second-law objective functions. The type of bottom surface in a DASC was shown to significantly affect its performance, particularly for relatively low particle loadings. Beyond a critical nanoparticle volume fraction value, collector performance was independent of bottom surface type and a DASC operates similar to a surface absorber. It has been also found that regardless of particle loading and bottom surface type, collectors with thinner nanofluid films always had a lower efficiency compared to collectors with thicker films. Water-based nanofluids were shown to offer stronger radiation absorption than therminol®-based ones up to a nanoparticle volume fraction of about 0.005%, at which level, therminol® becomes the stronger solar absorber. However, it was established that a nanofluid exhibiting stronger photo-thermal conversion does not necessarily lead to a higher collector efficiency. Finally, it was shown that optimizing with respect to a normalized combination of energy and exergy efficiencies (as opposed to only energy or exergy efficiencies) results in more reasonable design vectors with a balance between collector power and temperature gains.

Understanding the geologic evolution of Northern Tibetan Plateau with multiple thermochronometers

The early onset of deformation following the India-Asia collision, Neogene expanse of uplift, and complex systems that comprise strike-slip faults, thrust faults, and intermontane basins characterize the Cenozoic tectonism of Northern Tibetan Plateau and raise two prominent questions in orogenic geodynamics: 1) What mechanism(s) control(s) the transfer of stress related to the India-Asia collision across the distance of >2000km; and 2) Why the development of high topography was delayed in the Northern Tibetan Plateau and what does it reveal about how the internal forces and external boundary conditions evolved. To address these two questions, we reconstruct a holistic spatial-temporal deformation history of the Northern Tibetan Plateau by using a range of thermochronometers, with closure temperature spanning from 350°C to ~60–70°C. This multi-thermochronometer study reveals three stages of faulting related cooling, in the early Cretaceous, in Paleocene-Eocene and in middle-late Miocene. We observe that Paleocene-Eocene deformation was spatially restricted and mostly occurred on reactivated Cretaceous structures, indicating a control of pre-existing weakness on early Cenozoic deformation. Extensive Neogene deformation contrasts with restricted Paleocene-Eocene deformation and relatively quiescent shortening during the Oligocene-early Miocene, which implies a change in the regional tectonics regime. Global plate reconstructions show that this tectonic reorganization is coeval with an increase in Pacific-Asia plate convergence rates. We argue that this change in regional tectonics is a result of increasing constrictive environment of the eastern plate boundary, which changed the behavior of the Altyn Tagh fault—the boundary fault of Northern Tibetan Plateau, causing it to change from feeding slip into structures out of the plateau to feeding slip into structures at plateau margins.

A simple direct-forcing immersed boundary projection method with prediction-correction for fluid-solid interaction problems

In this paper, we propose a simple and novel direct-forcing immersed boundary (IB) projection method in conjunction with a prediction-correction (PC) process for simulating the dynamics of fluid-solid interaction problems, in which each immersed solid object can be stationary or moving in the fluid with a prescribed velocity. The method is mainly based on the introduction of a virtual force which is distributed only on the immersed solid bodies and appended to the fluid momentum equations to accommodate the internal boundary conditions at the immersed solid boundaries. More specifically, we first predict the virtual force on the immersed solid domain by using the difference between the prescribed solid velocity and the computed velocity, which is obtained by applying the Choi–Moin projection scheme to the incompressible Navier–Stokes equations on the entire domain including the portion occupied by the solid bodies. The predicted virtual force is then added to the fluid momentum equations as an additional forcing term and we employ the same projection scheme again to correct the velocity field, pressure and virtual force. Although this method is a two-stage approach, the computational cost of the correction stage is rather cheap, since the associated discrete linear systems need to be solved in the correction stage are same with that in the prediction stage, except the right-hand side data terms. Such a PC procedure can be iterated to form a more general method, if necessary. The current two-stage direct-forcing IB projection method has the advantage over traditional one-stage direct-forcing IB projection methods, consisting of the prediction step only, by allowing much larger time step, since traditional methods generally request quite small time step for flow field relaxed and adjusted to the solid body movement even using implicit scheme. Numerical experiments of several benchmark problems are performed to illustrate the simplicity and efficient performance of the newly proposed method. Convergence tests show that the accuracy of the velocity field is super-linear in space in all the 1-norm, 2-norm, and maximum norm. We also find that our numerical results are in very good agreement with the previous works in the literature and one correction at each time step appears to be good enough for the proposed PC procedure.

Finite Difference Method for Numerical Solution of Two Point Boundary Value Problems with Non-uniform Mesh and Internal Boundary Condition

Finite Difference Method for Numerical Solution of Two Point Boundary Value Problems with Non-uniform Mesh and Internal Boundary Condition

Generalization of Solovev’s approach to finding equilibrium solutions for axisymmetric plasmas with flow

Solovev’s approach of finding equilibrium solutions was found to be extremely useful for generating a library of linear-superposable equilibria for the purpose of shaping studies. This set of solutions was subsequently expanded to include the vacuum solutions of Zheng, Wootton and Solano, resulting in a set of functions {SOLOVEV_ZWS} that were usually used for all toroidally symmetric plasmas, commonly recognized as being able to accommodate any desired plasma shapes (complete-shaping capability). The possibility of extending the Solovev approach to toroidal equilibria with a general plasma flow is examined theoretically. We found that the only meaningful extension is to plasmas with a pure toroidal rotation and with a constant Mach number. We also show that the simplification ansatz made to the current profiles, which was the basis of the Solovev approach, should be applied more systematically to include an internal boundary condition at the magnetic axis; resulting in a modified and more useful set {SOLOVEV_ZWSm}. Explicit expressions of functions in this set are given for equilibria with a quasi-constant current density profile, with a toroidal flow at a constant Mach number and with specific heat capacity 1. The properties of {SOLOVEV_ZWSm} are studied analytically. Numerical examples of achievable equilibria are demonstrated. Although the shaping capability of the set {SOLOVE_ZWSm} is quite extensive, it nevertheless still does not have complete shaping capability, particularly for plasmas with negative curvature points on the plasma boundary such as the doublets or indented bean shaped tokamaks.

Finite Difference Method for Numerical Solution of Two Point Boundary Value Problems with Non-uniform Mesh and Internal Boundary Condition

Finite Difference Method for Numerical Solution of Two Point Boundary Value Problems with Non-uniform Mesh and Internal Boundary Condition

Parametric study on free vibration and instability of a functionally graded cracked shaft in a rotor-disc-bearing system: finite element approach

Free vibration and stability analysis are studied for a rotor-disk-bearing system having a radially functionally graded (FG) shaft with a transversely fully open crack, based on finite element (FE) approach. Both viscous and hysteretic internal damping are incorporated in the FE model of FG cracked shaft using two nodded Timoshenko beam element having four degrees of freedom (DOFs) at each node. Material properties of the FG cracked shaft are assumed temperature dependent and graded along radial direction following different material gradation law. FG shaft is made of two constituents material namely zirconia (ZrO2) and stainless steel (SS) where metallic (SS) contain is decreasing towards the outer diameter of the shaft. Extended Hamilton’s principle is employed to derive the system equations of motion (EOMs) of the FG cracked shaft system. A complete code is developed in MATLAB for correcting the formulation of modeling of crack and verified with existing published results. Influences of different material gradient index, temperature gradients, size and location of crack, viscous and hysteretic internal damping, slenderness ratio, and boundary condition on dynamic responses of the FG cracked shaft system are studied.

Vector extrapolation method for non-overlapping Schwarz iterations

In this paper, we propose a vector extrapolation method for accelerating the non-overlapping Schwarz iterations in the case of the non-linear reaction diffusion equations. The acceleration occurs at two levels: the acceleration of sequences produced by the fixed point algorithm, and the acceleration of sequences generated by Schwarz method. Specifically, the acceleration occurs in the internal Dirichlet boundary conditions. In order to illustrate the interest of the proposed algorithm, we have performed different test-cases of analytical examples, all results show the efficiency of the proposed approach in terms of CPU-Time, number of iterations and the rate of convergence.

Evaluating Hydraulic Properties of Biochar‐Amended Soil Aggregates by High‐Performance Pore‐Scale Simulations

Core Ideas SR‐mCT was used to characterize soil pore structures. Graphics processing unit–enhanced LBM was used to simulate the water flow. Effects of biochar amendment on soil hydraulic properties were evaluated. Biochar amendment can change the soil pore architecture and improve soil physical and hydraulic properties. In this study, two clayey soils were selected and treated with or without biochar amendment (four treatments). Synchrotron‐based X‐ray micro‐computed tomography (SR‐mCT) was used to characterize pore structures of the selected aggregates from the treatments. Based on the high‐resolution soil pore structure information, the three‐dimensional lattice Boltzmann (LB) method was used to simulate water flow in pores. The high‐resolution soil pore structures were also used to define the internal boundary conditions in the LB method that was accelerated by graphics processing unit parallel computing. Results showed that biochar amendment increased soil permeability by at least one order of magnitude, reduced tortuosity by 20 to 30%, and resulted in more uniformly distributed pore water velocities, which was attributed to the change in soil pore geometry. This study is the first physically based modeling effort to assess the effect of biochar amendment on hydraulic properties of soil aggregates. The SR‐mCT method and modeling approach developed in the research play an important role in understanding the mechanisms of water and nutrient transport in soil at the pore scale.

Analytical solution of Fourier and non-Fourier heat transfer in longitudinal fin with internal heat generation and periodic boundary condition

In this paper, the analytical solution of Fourier and non-Fourier model of heat transfer in longitudinal fin in presence of internal heat generation has been studied under periodic boundary condition. The whole analysis is given in dimensionless form. These two mathematical models have been solved analytically, using Laplace transform technique. Temperature distribution in longitudinal fin is measured using residual theorem in complex plane for the inverse Laplace transform technique. Thermal wave nature is appeared for small value of Fo. The longitudinal fin temperature is evaluated for different value of parameters with respect to space coordinate. The effect of variability of different parameters on temperature distribution in fin is studied in detailed. It has been observed that the cooling process is faster in non-Fourier model in comparison to Fourier model.

A finite difference method for the numerical solving general third order boundary-value problem with an internal boundary condition

In this paper we have proposed a finite difference method for the numerical solving general third order boundary-value problem. Under appropriate conditions we have discussed the convergence of the proposed method. The computational results in experiment on test problems verify the efficiency and theoretically established second order accuracy of the proposed method.