An Accurate and Efficient Stochastic Solver for Transient Thermal Analysis With Mixed Boundary Conditions

Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.

Treatment of mixed and nonuniform boundary conditions in GDQ vibration analysis of rectangular plates

Analysis of layered panels with mixed edge boundary conditions using state space differential quadrature method

Using Burgers' equation with mixed Neumann-Dirichlet boundary conditions, we highlight a problem that can arise in the numerical approximation of nonlinear dynamical systems on computers with a finite precision floating point number system. We describe the dynamical system generated by Burgers' equation with mixed boundary conditions, summarize some of its properties and analyze the equilibrium states for finite dimensional dynamical systems that are generated by numerical approxima- tions of this system. It is important to note that there are two fundamental differences between Burgers' equation with mixed Neumann-Dirichlet boundary conditions and Burgers' equation with both Dirich- let boundary conditions. First, Burgers' equation with homogenous mixed boundary conditions on a finite interval cannot be linearized by the Cole-Hopf transformation. Thus, on finite intervals Burgers' equation with a homogenous Neumann boundary condition is truly nonlinear. Second, the nonlinear term in Burgers' equation with a homogenous Neumann boundary condition is not conservative. This structure plays a key role in understanding the complex dynamics generated by Burgers' equation with a Neumann boundary condition and how this structure impacts numerical approximations. The key point is that, regardless of the particular numerical scheme, finite precision arithmetic will always lead to numerically generated equilibrium states that do not correspond to equilibrium states of the Burgers' equation. In this paper we establish the existence and stability properties of these numerical stationary solutions and employ a bifurcation analysis to provide a detailed mathematical explanation of why numerical schemes fail to capture the correct asymptotic dynamics. We extend the results in (E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, Math. Comput. Modelling 35 (2002) 1165-1195) and prove that the effect of finite precision arithmetic persists in generating a nonzero numerical false solution to the stationary Burgers' problem. Thus, we show that the results obtained in (E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, Math. Comput. Modelling 35 (2002) 1165-1195) are not dependent on a specific time marching scheme, but are generic to all convergent numerical approximations of Burgers' equation.

High-resolution finite element models of trabecular bone can be used to study trabecular structure-function relationships, elasticity, multiaxial strength, and tissue remodelling in more detail than experiments. Beside effects of the model size, scan/analysis resolution, segmentation process, etc., the type of the applied boundary conditions (BCs) have a strong influence on the predicted elastic properties. Appropriate BCs have to be applied on hexahedral digital finite element models in order to obtain effective elastic properties. Homogeneous displacement BCs as proposed by Van Rietbergen et al. (J Biomech 29(12):1653-1657, 1996) lead to "apparent" rather than to "effective" elastic properties. This study provides some answers concerning such differences by comparing various BC types (uniform displacement, mixed BCs, periodic BCs), different volume element definitions (original and mirrored models), and several bone volume fractions (BVTV ranging from 6.5 to 37.6%). First, the mixed BCs formulated by Hazanov (Arch Appl Mech 68(6):385-394, 1998) are theoretically extended to shear loading of a porous media. Second, six human bone samples are analyzed, their orthotropic Young's moduli, shear moduli, and Poisson's ratios computed and compared. It is found that the proposed mixed BCs give exactly the same effective elastic properties as periodic BCs if a periodic and orthotropic micro-structured material is used and thus denoted as "periodicity compatible" mixed uniform BCs (PMUBCs). As bone samples were shown to be nearly orthotropic for volume element side lengths > or =5 mm the proposed mixed BCs turn out to be the best choice because they give again essentially the same overall elastic properties as periodic BCs. For bone samples of smaller dimensions ( < 5 mm) with a strong anisotropy (beyond orthotropy) uniform displacement BCs remain applicable but they can significantly overestimate the effective stiffness.

Boundary conditions have a vital role in the numerical solution of the partial differential equations governing fluid flows, and they have a great influence over the numerical stability and accuracy of the final solutions. One of the most important physical boundary conditions in flow field analysis, especially for inviscid flows, is the wall boundary condition imposed on body surfaces. To solve a three-dimensional compressible Euler equation (with five PDE's), a total of five boundary conditions on the body surface should be prescribed. The wall's velocity magnitude is one of the parameters to be determined, and the way this velocity magnitude is calculated affects the accuracy and stability of the numerical approach. In this paper, four different methods for calculation of the wall velocity magnitude are introduced, tested and compared against several test cases of subsonic and supersonic flows. Since there are many problems where both subsonic and supersonic flows coexist, a mixed boundary condition takes into account the flow Mach number is proposed. The mixed boundary condition is applied to several test cases and the stability of the method is examined.

The differences between the interdecadal variability under mixed and constant flux boundary conditions are investigated using a coarse-resolution ocean model in an idealized flat-bottom single-hemisphere basin. Objective features are determined that allow one type of oscillation to be distinguished versus the other. First, by performing a linear stability analysis of the steady state obtained under restoring boundary conditions, it is shown that the interdecadal variability under constant flux and mixed boundary conditions arises, respectively, from the instability of a linear mode around the mean stratification and circulation and from departure from the initial state. Based on the budgets of density variance, it is shown next that the two types of oscillations have different energy sources: Under the constant-flux boundary condition (the thermal mode), the downgradient meridional eddy heat flux in the western boundary current regions sustains interdecadal variability, whereas under mixed boundary conditions (the salinity mode), a positive feedback between convective adjustment and restoring surface heat flux is at the heart of the existence of the decadal oscillation. Furthermore, the positive correlations between temperature and salinity anomalies in the forcing layer are shown to dominate the forcing of density variance. In addition, the vertical structure of perturbations reveals vertical phase lags at different depths in all tracer fields under constant flux, while under mixed boundary conditions only the temperature anomalies show a strong dipolar structure. The authors propose that these differences will allow one to identify which type of oscillation, if any, is at play in the more exhaustive climate models.

The influence of the mixed mechanical boundary condition, defined by a linear relation between velocity at the boundary and stress, on the non-linear stability of steady states and on the localization of the deformation is investigated by numerical means. This mixed boundary condition is known to prevail during a torsional Kolsky bar test and depends on the geometry of the incident bar and of the specimen. It is shown that the coefficient which enters the mixed condition governs the evolution of the flow from a perturbed, unstable steady state towards a stable attractor. The same coefficient is found to have little influence on the nominal strain at initiation of the localization which appears to be governed by the total energy dissipated. However, the rate at which the stress drops during the development of the localization is affected by the mixed boundary condition.

We show error estimates for a cut finite element approximation of a second order elliptic problem with mixed boundary conditions. The error estimates are of low regularity type where we consider the case when the exact solution u ∈ H s u \in H^s with s ∈ ( 1 , 3 / 2 ] s\in (1,3/2] . For Nitsche type methods this case requires special handling of the terms involving the normal flux of the exact solution at the the boundary. For Dirichlet boundary conditions the estimates are optimal, whereas in the case of mixed Dirichlet-Neumann boundary conditions they are suboptimal by a logarithmic factor.

The problem of scattering of an arbitrarily directed scalar incident plane wave by a fundamental surfacE, of orthogonality with mixed linear boundary conditions is solved for arbitrary separable coordinate system. From the Fredholm integral equation of the second kind, derived from the Helmholtz solution, a simple linear relationship is found between the unknown coefficients in the expansions of the wave potential function and its normal derivative on the fundamental surface of orthogonality, in terms of the eigenfunctions of the separable coordinate system. By applying the mixed linear boundary conditions on the fundamental surface of orthogonality another set of simple linear relationships between the unknown coefficients is found. From the two sets of linear equations, the unknown coefficients may be determined and the scattered field calculated. The above method may be applied to both Laplace and Helmholtz equations. Since we require for the solution only the free space Green’s function expansion, the problem of finding explicit representations of the various specialized Green’s functions is avoided.

A general mixed kinetic-diffusion boundary condition is formulated to account for the out-of-equilibrium kinetics in the Knudsen layer. The mixed boundary condition is used to investigate the problem of quasi-steady evaporation of a droplet in an infinite domain containing inert gases. The widely adopted local thermodynamic equilibrium assumption is found to be the limiting case of infinitely large kinetic Péclet number ${{Pe}_k}$ , and it introduces significant error for ${{Pe}_k} \leqslant O(10)$ , which corresponds to a typical droplet radius $a$ of a few micrometres or smaller. When compared with experimental data, solutions based on the mixed boundary condition, which take into account the temperature jump across the Knudsen layer, better predict the time evolution of $a$ than the classical $D^2$ -law (i.e. $a^2 \propto t$ , where $t$ denotes time). In the slow evaporation limit, an analytical solution is obtained by linearising the full formulation about the equilibrium condition, which shows that the $D^2$ -law can be recovered only in the large ${{Pe}_k}$ limit. For small ${{Pe}_k}$ , where the process is dominated by kinetics, a linear relation, i.e. $a \propto t$ , emerges. When the gas phase density approaches the liquid density (e.g. at high-pressure or low-temperature conditions), the increase in the chemical potential of the liquid phase due to the presence of inert gases needs to be accounted for when formulating the mixed boundary condition, an effect largely ignored in the literature so far.

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.

We give a general analysis of AdS boundary conditions for spin-$3/2$ Rarita-Schwinger fields and investigate boundary conditions preserving supersymmetry for a graviton multiplet in ${\mathrm{AdS}}_{4}$. Linear Rarita-Schwinger fields in ${\mathrm{AdS}}_{d}$ are shown to admit mixed Dirichlet-Neumann boundary conditions when their mass is in the range $0\ensuremath{\le}|m|&lt;1/2{l}_{\mathrm{AdS}}$. We also demonstrate that mixed boundary conditions are allowed for larger masses when the inner product is ``renormalized'' accordingly with the action. We then use the results obtained for $|m|=1/{l}_{\mathrm{AdS}}$ to explore supersymmetric boundary conditions for $\mathcal{N}=1$ ${\mathrm{AdS}}_{4}$ supergravity in which the metric and Rarita-Schwinger fields are fluctuating at the boundary. We classify boundary conditions that preserve boundary supersymmetry or superconformal symmetry. Under the AdS/CFT dictionary, Neumann boundary conditions in $d=4$ supergravity correspond to gauging the superconformal group of the three-dimensional CFT describing M2-branes, while $\mathcal{N}=1$ supersymmetric mixed boundary conditions couple the CFT to $\mathcal{N}=1$ superconformal topologically massive gravity.

A Laplace-transform boundary element model for pumping tests in irregularly shaped double-porosity aquifers

Using the adjoint of a fully three-dimensional primitive equation ocean model in an idealized geometry, spatial variations in the sensitivity to surface boundary forcing of the meridional overturning circulation’s strength are studied. Steady-state sensitivities to diapycnal mixing, wind stress, freshwater, and heat forcing are examined. Three different, commonly used, boundary-forcing scenarios are studied, both with and without wind forcing. Almost identical circulation is achieved in each scenario, but the sensitivity patterns show major (quantitative and qualitative) differences. Sensitivities to surface forcing and diapycnal mixing are substantially larger under mixed boundary conditions, in which fluxes of freshwater and heat are supplemented by a temperature relaxation term or under flux boundary conditions, in which climatological fluxes alone drive the circulation, than under restoring boundary conditions. The sensitivity pattern to diapycnal mixing, which peaks in the Tropics is similar both with and without wind forcing. Wind does, however, increase the sensitivity to diapycnal mixing in the regions of Ekman upwelling and decreases it in the regions of Ekman downwelling. Wind stress in the Southern Oceans plays a crucial role in restoring boundary conditions, but the effect is largely absent under mixed or flux boundary conditions. The results highlight how critical a careful formulation of the surface forcing terms is to ensuring a proper response to changes in forcing in ocean models.