Compatible Boundary Research Articles

Entropy generation in Casson nanofluid flow over a lubricated wall in the presence of induced magnetic field and activation energy

ABSTRACT The research on entropy generation in a mechanical system is quite imperative owing to its immense applications in the production of advanced cooling devices and medical appliances. The basic need on this research is how to optimize the irreversibilities within the systems and channelize that energy towards the benefit. This study elucidates the entropy generation analysis of a Casson nanofluid near a stagnation point over a lubricated vertical wall. A thin film of shear-thinning liquid with power law index of 0.5 is employed on the wall to ensure the lubrication. The effect of induced magnetic field and the chemical reaction stimulated by the activation energy is also incorporated within the flow. The non-linear PDEs of the substantive problem along with the compatible interfacial and boundary conditions are reverted into ODEs with the aid of suitable similarity transformation, and then cracked numerically using the sixth-order Runge Kutta method assisted with Nachtsheim–Swigert shooting practice. One of the significant findings reveals that the fluid friction irreversibilities in the entropy generation are diminished with the magnetic parameter and the Casson parameter, whereas the thermal irreversibility predominates over the total entropy generation for the escalating values of the activation energy parameter.

Fourth-Order Accurate Compact Scheme for First-Order Maxwell’s Equations

We construct a compact fourth-order scheme, in space and time, for the time-dependent Maxwell’s equations given as a first-order system on a staggered (Yee) grid. At each time step, we update the fields by solving positive definite second-order elliptic equations. We develop compatible boundary conditions for these elliptic equations while maintaining a compact stencil. The proposed scheme is compared computationally with a non-compact scheme and with a convolutional dispersion relation preserving (DRP) scheme.

DC3DPAFEM: An efficient and accurate 3-D direct current resistivity anisotropic forward modeling software for complex geological settings

Nowadays, there is a growing trend that direct current (DC) field surveys are shifting towards challenging areas characterized by mountainous topography and electrical anisotropy. Given these complex geological settings, there is an urgent need for 3-D DC forward modeling software capable of effectively addressing large-scale problems and delivering accurate modeling results to interpret field data. However, most open-source software packages face certain limitations, such as the high numerical cost to handle complex surface topography, the lack of consideration for anisotropic conductivity, the absence of mesh refinement techniques to guarantee accuracy in forward modeling, and the lack of parallel computing techniques to solve large-scale problems. In this study, we develop an efficient and highly accurate 3-D DC anisotropic forward modeling software, namely DC3DPAFEM, using the adaptive finite element algorithm based on the unstructured tetrahedral mesh. Firstly, we construct a strong compatible boundary value problem (BVP) for 3-D anisotropic DC problems by adopting a specialized secondary potential approach to handle the surface topography efficiently. Then, we develop a goal-oriented adaptive mesh refinement (AMR) technique to ensure accurate forward modeling results, even with a coarse initial mesh. To ensure time and memory efficiency, we employ a robust conjugate gradient (CG) algorithm preconditioned by the algebraic multigrid (AMG) solver to solve the large-scale linear system of equations resulting from complex geological structures. We aim to investigate the performance of the AMG scheme in anisotropic DC cases. Furthermore, we incorporate the domain decomposition technique into the iterative solution scheme for further efficiency gains. This technique significantly improves computing efficiency for large-scale problems in parallel clusters. Finally, we conduct comprehensive performance tests for DC3DPAFEM using a two-layer anisotropic model and a 3-D complex model with undulating terrain. The results of both examples validate the accuracy of DC3DPAFEM, as they closely align with the analytical solutions and the solutions obtained from the existing 3-D DC forward modeling code. Compared to traditional direct solver MUMPS and ILU-preconditioned iterative solvers, DC3DPAFEM exhibits highly scalable performance for large-scale problems, offering significant advantages in terms of memory and time consumption. Overall, DC3DPAFEM demonstrates substantial advances in efficiency, accuracy, and practicality through a series of numerical examples. This open-source code provides an efficient and available tool for developing a 3-D DC inversion method that can deal with large-scale problems involving intricate topography and anisotropic media.

A compatible boundary condition-based topology optimization paradigm for static mechanical cloak design

We propose a novel compatible boundary condition-based topology optimization formulation for realizing static mechanical cloaking of linear elastic structures. The key idea is to make the actual boundary conditions along the exterior of the cloaking region consistent with the reference boundary conditions, including the displacement and the nodal force, by designing the cloak structure solely. This compatible boundary condition-based optimization paradigm is not only applicable to various shaped voids (e.g., circle, ellipse, star, Mickey, square, “DUT” logo) and diverse loading conditions (e.g., pressure-sliding, biaxial-stretching, linear-stretching, random-stretching, stretching-sliding) but also demonstrates insensitivity to element quantities and slight variations of external load, ensuring consistent and robust elastostatic cloaking performance. Additionally, adopting the Moving Morphable Voids (MMV) method guarantees the optimized structures have the merits of explicit geometric boundaries and easy fabrication. Numerical examples and experimental validation demonstrate the effectiveness of the proposed approach. The present approach can be easily extended for cloak design in three dimensions and multiphysics.

Irreversibility of heat transfer with the curvature effect on peristaltic thrusting of micropolar fluid in a resilient channel in the presence of heat generation: nonlinear analysis

In this work, we perform an investigation of the peristaltic thrusting of an incompressible viscous micropolar fluid in a symmetric resilient channel with consideration of streamline curvature effects and inertia forces. We obtained the partial differential equations for fluid flow in two dimensions for micropolar fluids in addition to the heat equation. These equations have been transformed to another domain by using a transformation between the fixed and wave frames. Asymptotic solutions of have been obtained by using the compatible boundary conditions using the perturbation technique of arbitrary Reynolds number and long-drawn wavelength. The solutions for longitudinal, normal and spin velocities, stream function, heat transfer and rate of working walls have been obtained. The several of biophysical parameters influence has been demonstrated through the graphs. Both time flow rate and pressure gradient relations are obtained. Different characteristics for fluid flow, temperature profile, normal force, microrotation (spin) velocity and Nusslet number are greatly influenced at the channel walls by the presence of micropolar fluid. Trapping and pumping phenomena are further discussed. It is found that the maximum value for the Bejan number occurs near the centre of the channel, i.e. the heat transfer irreversibility dominates around the centre of the channel.

Fractional-step finite difference schemes for incompressible elasticity on overset grids

Efficient finite-difference schemes for the numerical solution of the time-dependent equations of incompressible linear elasticity on complex geometry are presented. The schemes are second-order accurate in space and time, and solve the equations in displacement-pressure form. The algorithms use a fractional-step approach in which the time-step of the displacements is performed separately from the solution of a Poisson problem to update the pressure. Complex geometry is treated with curvilinear overset grids. Compatibility boundary conditions and an upwind dissipation for wave equations in second-order form are included to ensure stability for overset grids, and for the difficult case of traction (free-surface) boundary conditions. A divergence damping term is added to keep the dilatation small. The stability of the schemes is studied with GKS mode analysis. Exact eigen-mode solutions are obtained for several problems involving rectangular, cylindrical and spherical configurations, and these are valuable as benchmark problems. The accuracy and stability of the approach in two and three space dimensions is illustrated by comparing the numerical solutions with the exact solutions of the benchmark problems.

On the dynamics of nanoscale structures upon a novel mixture model of elasticity: buckling analysis, vibration analysis, and wave propagation

In this study, a novel mixture model of elasticity, that can cover all size-dependent theories together with simultaneously taking of softening and stiffening impacts, is regarded as mathematical modeling and dynamic analysis of Timoshenko nanobeams. Since the mixture model is the combination of Eringen's two-phase theory with two-phase stress-driven theory, therefore, there are four parameters containing two various types of nonlocal parameters and two additional local phase fraction factors. In this work, after introducing the mixture model together with essential assumptions and relations, the integral form of relations with an equivalent differential form of ones is illustrated. Next, essential conditions are considered and satisfied to match the integral relations with differential relations without any conflict of restrictions are obtained. Finally, through some examples, the essential relations together with compatibility boundary conditions are derived to analyze the buckling, vibration, and wave propagation of Timoshenko nanobeams. Utilizing an efficient numerical solution, the results are achieved, and the validity as well as integrity of the constitutive differential relations are examined through some validation studies. The output results disclose that the previous results in the field of contemporary continuum theories are the particular state of dynamic response of the nanobeams.

Stable, entropy-pressure compatible subsonic Riemann boundary condition for embedded DG compressible flow simulations

One approach to reducing the computational cost of simulating transitional compressible boundary layer flow is to adopt a near body reduced domain with boundary conditions enforced to be compatible with a computationally cheaper three-dimensional RANS simulation. In such an approach it is desirable to enforce a consistent pressure distribution which is not typically the case when using the standard Riemann inflow boundary conditions. We therefore revisit the Riemann problem adopted in many DG based high fidelity formulations.Through analysis of the one-dimensional linearized Euler equations it is demonstrated that maintaining entropy compatibility with the RANS simulation is important for a stable solution. It is also necessary to maintain the invariant for the Riemann outflow boundary condition in a subsonic flow leaving one condition that can be imposed at the inflow boundary. Therefore the entropy-pressure enforcement is the only stable boundary condition to enforce a known pressure distribution. We further demonstrate that all the entropy compatible inflow Riemann boundary conditions are stable providing the invariant compatible Riemann outflow boundary condition is also respected.Although the entropy-pressure compatible Riemann inflow boundary condition is stable from the one-dimensional analysis, two-dimensional tests highlight divergence in the inviscid problem and neutrally stable wiggles in the velocity fields in viscous simulations around the stagnation point. A two-dimensional analysis about a non-uniform baseflow assumption provides insight into this stability issue (ill-posedness) and motivates the use of a mix of inflow boundary conditions in this region of the flow.As a validation we apply the proposed boundary conditions to a reduced domain of a wing section normal to the leading-edge of the CRM-NLF model taken out of a full three-dimensional RANS simulation at Mach 0.86 and a Reynolds number of 8.5 million. The results show that the entropy-pressure compatible Riemann inflow boundary condition leads to a good agreement in pressure distribution.

Local Compatibility Boundary Conditions for High-Order Accurate Finite-Difference Approximations of PDEs

We describe a new approach to derive numerical approximations of boundary conditions for high-order accurate finite-difference approximations. The approach, called the Local Compatibility Boundary Condition (LCBC) method, uses boundary conditions and compatibility boundary conditions derived from the governing equations, as well as interior and boundary grid values, to construct a local polynomial, whose degree matches the order of accuracy of the interior scheme, centered at each boundary point. The local polynomial is then used to derive a discrete formula for each ghost point in terms of the data. This approach leads to centered approximations that are generally more accurate and stable than one-sided approximations. Moreover, the stencil approximations are local since they do not couple to neighboring ghost-point values which can occur with traditional compatibility conditions. The local polynomial is derived using continuous operators and derivatives which enables the automatic construction of stencil approximations at different orders of accuracy. The LCBC method is developed here for problems governed by second-order partial differential equations, and it is verified for a wide range of sample problems, both time-dependent and time-independent, in two space dimensions and for schemes up to sixth-order accuracy.

Three-dimensional rotating flow of Cu–Al2O3/kerosene oil hybrid nanofluid in presence of activation energy and thermal radiation

The present exploration talks to a steady and incompressible chemically reacting hybrid-nanofluid rotating flow toward a stretchable sheet in presence of activation energy and magnetic field. We have chosen kerosene oil as base fluid and two nanoparticles specifically Al2O3 and Cu for hybrid-nanofluid and Al2O3 for regular nanofluid. Relative studies of imprint factors between hybrid nanofluid and customary nanofluid have been implemented for our present article. The leading system of PDEs subject to their compatible boundary conditions are renovated into a system of ODEs and then resolved by the Runge–Kutta–Fehlberg quadrature method with shooting practice. The stimulus of ignite flow parameters on the flow precise is carried off accurately through figures, bar diagrams and tables. Our analysis conveys that the primary velocity profiles for both hybrid and regular nanofluid diminish with intensifying values of Hartmann number and rotational parameter, however secondary velocity profile enhances with augmentation of the rotational parameter. Temperature profiles of both category fluids lead to increase with growing values of radiation parameter and Hartmann number, whereas opposite phenomena observe for concentration profiles with improving values of chemical reaction parameter. Again, rate of heat transfer diminishes sharply for leading Hartmann number.

Two-dimensional Darcy–Bénard convection evolving in Fourier space

Nonlinear transient convection in a porous rectangle heated from below is studied by an analytically based method. Fourier series for the temperature and streamfunction are applied, where each Fourier coefficient evolves in time according to a coupled set of ordinary differential equations. The mathematical method can be considered as a recursive nonlinear mapping in Fourier space from a given state at a time t to an updated state at time t + dt, with an infinitesimal time increment. This nonlinear evolution in Fourier space requires normal-mode compatible boundary conditions along the entire boundary. Each Fourier coefficient gets three contributions during its updating in time: (i) one decay term due to thermal diffusion, governed by linear theory; (ii) one growth term due to buoyancy, governed by linear theory; and (iii) quadratic nonlinearities from the convection term in the heat equation, involving all pairwise interactions between the Fourier modes. We present numerical computations with a standard Runge–Kutta method, with Rayleigh numbers up to ten times the well-known critical value 4π2. Our plots are produced with Fourier series truncated to include 15 normal modes in both the horizontal and vertical directions. For validation of our method, we present tables for the Nusselt number of steady convection, with a higher number (up to 20) of normal modes included in the truncated system of equations. Our computations of transient nonlinear convection lead to steady states. A final steady state is not unique for a given geometry, but depends on the initial state and the Rayleigh number. This ambiguity of steady states is depicted by a hysteresis loop. The Malkus hypothesis of maximal heat transfer is put into perspective. This hypothesis does not pick a preferred cell width, but it nevertheless constrains the hysteresis loop.

Performance assessment of internal porous structures on liquid sloshing in various 3D tanks by multi-domain IGABEM

An isogeometric boundary element method (IGABEM) is extended in this work to study liquid sloshing characteristics in various 3D tanks with arbitrary internal bottom-mounted porous structures. Unlike the classical boundary element method (BEM), the present method uses the non-uniform rational B-splines(NURBS) instead of the piecewise Lagrange polynomials as the shape function to approximate both the domain boundary and field variables, it completely inherits the advantages of the traditional BEM and the isogeometric analysis (IGA), such as the properties of only boundary discretization needing, higher order continuity, self-adaptability and so on. All the information required in the IGABEM for the mesh generation can be obtained from the CAD software, which may evidently reduce the time and memory consumptions of preprocessing, meanwhile, since the same basis functions (NURBS) are used for both the IGABEM and CAD models, the proposed method can exactly reconstruct the geometry of the analysis domain without any error, and this substantively contributes to the high calculation accuracy of IGABEM. In this paper, by using a zoning method the entire domains of fluid are divided into several sub-domains with the consideration of compatibility boundary conditions besides the porous structures. Owning to the constant cross section of the container, impermeable boundary condition at the tank bottom and the linear boundary condition at the free surface, the 3D Laplace equation (which governs the 3D sloshing problem) is transformed into a couple of Helmholtz equation and modified Helmholtz equations (corresponding to the propagating and evanescent modes, respectively). Green's theorem and the divergence theorem of Gauss are used in this paper to derive the IGABEM system equations for the two types of equations. Moreover, by introducing a series of eigen-functions associated with the vertical coordinate, the potential and velocity boundary conditions are expanded into those associated with the propagating and evanescent modes, which can be applied upon the Helmholtz equation and modified Helmholtz equations. Finally, the present problem can be solved by solving these equations and combining the results linearly. Accuracy and convergence of the present IGABEM technique are verified through numerical tests by comparing the obtained results with analytical, experimental solutions or those calculated with other numerical method, thereafter, liquid sloshing problems in the cubic or circular cylindrical vessels with the coaxial or eccentric elliptical and circular cylindrical porous structures are analyzed. The main influences of the geometric parameter, porous-effect parameter, number and arrangement of the porous structures are investigated in detail, some conclusions are outlined in the end of this paper.

Entropy generation analysis on convective radiative stagnation point flow of a Casson nanofluid in a non-Darcy porous medium with induced magnetic field and activation energy: the multiple quadratic regression model

The present article analyzes the entropy generation on the nonlinear thermally radiating magnetohydrodynamic Casson nanofluid flow closer to the stagnation point inside a non-Darcy porous medium with viscous dissipation, magnetic dissipation, mixed convection, and heat generation. The induced magnetic field's impact and the implications of the chemical reaction and the activation energy are also included in the study. A mathematical model based on the substantive problem is developed, consisting of nonlinear partial differential equations and compatible boundary conditions. The mathematical model is converted to a set of nonlinear ordinary differential equations with appropriate boundary conditions using a suitable similarity transformation, which is then solved using the spectral quasi-linearization method. A complete examination is executed for achieving in-depth information of different varying flow parameters' consequences on the flow field and the significant physical quantities using several graphs and tables. Using the regression models on the reduced Nusselt number NuxRex−1/2 and the reduced Sherwood number ShxRex−1/2, the maximum relative errors are shown as 0.15%–0.30% and 0.016024%–0.016225%, respectively. The present results signify that the larger velocity ratio parameter boosts the velocity along the x-axis and the transverse velocity for M = 3, but the transverse velocity for M = 1, temperature, and induced magnetic field follow a reverse pattern.

Singular boundary conditions for Sturm–Liouville operators via perturbation theory

AbstractWe show that all self-adjoint extensions of semibounded Sturm–Liouville operators with limit-circle endpoint(s) can be obtained via an additive singular form-bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in \{1,2\}$ . This characterization generalizes the well-known analog for semibounded Sturm–Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as $$ \begin{align*} \boldsymbol{A}_{ {}_{\scriptstyle \Theta}}=\boldsymbol{A}_0+\mathbf{B}\Theta\mathbf{B}^*, \end{align*} $$ where $\boldsymbol {A}_0$ is a distinguished self-adjoint extension and $\Theta $ is a self-adjoint linear relation in $\mathbb {C}^d$ . The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form-bounded with respect to $\boldsymbol {A}_0$ , i.e., it belongs to $\mathcal {H}_{-1}(\boldsymbol {A}_0)$ , with possible “infinite coupling.” A boundary triple and compatible boundary pair for the symmetric operator are constructed to ensure that the perturbation is well defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations $\Theta $ .The merging of boundary triples with perturbation theory provides a more holistic view of the operator’s matrix-valued spectral measures: identifying not just the location of the spectrum, but also certain directional information.As an example, self-adjoint extensions of the classical Jacobi differential equation (which has two limit-circle endpoints) are obtained, and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.

Recrystallization behavior and grain boundary character evolution in Co-Cr alloy from selective laser melting to heat treatment

Co-Cr alloy fabricated by selective laser melting was used to investigate the recrystallization behavior and grain boundary character evolution from As-SLM to isothermal annealed at 1200 °C for 5 min to 60 min. The As-SLM Co-Cr alloy has a prominent residual strain, high densities of dislocations and stacking faults, and solute segregation along the subgrain boundaries. The driving force for recrystallization of As-SLM Co-Cr alloy is mainly provided by phase transformation rather than crystal defects, which also offer plenty of routeways for both low- and high-angle grain boundary diffusion. In addition, the stacking faults in As-SLM Co-Cr alloy can generate morphologically complex twins by extending and gliding. The obtained optimal Co-Cr alloy has a large fraction (more than 80%) of special boundaries (Σ3, Σ9, and Σ27) after complete recrystallization, which enables high designing freedom and near-net-shape production towards additive manufacturing (AM) compatible grain boundary engineering (GBE) processing strategy.

Multi‐patch variational differential quadrature method for shear‐deformable strain gradient plates

AbstractThe integration of generalized differential quadrature techniques and finite element (FE) methods has been developed during the past decade for engineering problems within classical continuum theories. Hence, the main objective of the present study is to propose a novel numerical strategy called the multi‐patch variational differential quadrature (VDQ) method to model the structural behavior of plate structures obeying the shear deformation plate theory within the strain gradient elasticity theory. The idea is to divide the two‐dimensional solution domain of the plate model into sub‐domains, called patches, and then to apply the VDQ method along with the FE mapping technique for each patch. The formulation is presented in a weak form and due to the ‐continuity requirements the corresponding compatibility conditions are applied through the patch interfaces. The Lagrange multiplier technique and the penalty method are implemented to apply the higher‐order compatibility conditions and boundary conditions, respectively. To show the efficiency of the proposed method, numerical results are provided for plate structures with both regular and irregular solution domains. The provided numerical examples demonstrate the applicability and accuracy of the method in predicting the bending and vibration behavior of plate structures following the higher‐order plate model.

A novel computational method of the free energy for an Ising model on Cayley tree of order three

In this paper, we consider the Ising-Vannimenus model on a Cayley tree of order three. We provide an efficient approach to give some explicit formulas for the free energies and the entropies corresponding to the boundary conditions (b.c.) for Ising-Vannimenus model including compatible and translation-invariant boundary conditions.

Transient nonlinear Rayleigh–Bénard convection with single-mode initiation

Transient nonlinear convection in a finite Prandtl number fluid heated uniformly from below is studied analytically and numerically. We consider the simple geometry of a square cavity with normal-mode compatible boundary conditions. By design, only the marginal state of convection onset contributes to the initial condition for the two-dimensional supercritical convection. The thermal amplitude and the flow amplitude are taken as two independent initial conditions. There are two ways of initiation: (i) soft start with a very small initial amplitude, leading to early exponential growth. (ii) Kick-started transient convection with relatively large initial amplitudes, by which we perform a small-time expansion for benchmarking. Seemingly complicated transient flow occurs with a kick-start where the initial spin and the initial buoyancy torque are in conflict. However, the intricate spiraling flow decays after a couple of reversals of flow directions, and a steady convection settles. This is due to the strict antisymmetry of the temperature perturbation around the mid-point of the cavity, in combination with the symmetry of the flow field.

Energy-production-rate preserving numerical approximations to network generating partial differential equations

We recast a network generating partial differential equation system into a singular limit of a dissipative gradient flow model, which not only identifies the consistent physical boundary conditions but also generates networks. We then develop a set of structure-preserving numerical algorithms for the gradient flow model. Using the energy quadratization (EQ) method, we reformulate the gradient flow system into an equivalent one with a quadratic energy density by introducing auxiliary variables. Subsequently, we devise a series of fully discrete, linear, second order, energy-production-rate preserving, finite difference algorithms to solve the EQ-reformulated PDE system subject to various compatible boundary conditions. We show that the numerical schemes are energy-production-rate preserving for any time steps. Numerical convergence tests are given to validate the accuracy of the fully discrete schemes. Several 2D numerical examples are given to demonstrate the capability of the schemes in predicting network generating phenomena with the gradient flow PDE system, especially, the original network generating PDE model.

Transient free convective flow through a vertical cylinder filled with a porous material

AbstractThis article aims to explore the impressive impact of emerging parameters on transient fully evolved free convective flow inside a vertical cylinder containing a porous material. The mathematical formulation of the model related to the considered physical circumstance is presented under compatible boundary conditions. Closed‐form solutions are received for the velocity field, the temperature distribution, mass flux, skin friction, and the Nusselt number in terms of Bessel functions and modified Bessel functions of the first kind. Impressive effects of parameters such as the Darcy number , Prandtl number , viscosity ratio , and also time on both the velocity and temperature distribution have been explored employing graphs and tables. It is irradiated by analysis that flow erection, heat transfer rate, skin friction, and mass flux are admirably impacted by the Prandtl number, the Darcy number, viscosity ratio parameter, and time. It is found that both the velocity and temperature field profiles rise with the rising value of time and ultimately attain their steady state. Moreover, the Prandtl number and the viscosity ratio parameter reduce the velocity profiles, while the reverse phenomenon occurs with the Darcy number.