Boundary conditions (Differential equations)

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Modelling Boundary Conditions in High-Order, Nodal, Time-Domain Finite Element Methods

Modelling Boundary Conditions in High-Order, Nodal, Time-Domain Finite Element Methods

Abstract Accurate modeling of boundary conditions is an important aspect in room acoustic simulations. It has been shown that the acoustics of rooms is not only dependent on the frequency characteristic of the complex bound- ary impedance, but also on the angle dependent properties of the impedance (“extended reaction”). This pa- per presents a computationally efficient method for modeling local-reaction (LR) and extended-reaction (ER) boundary conditions in high-order, nodal, time-domain finite element methods, such as the spectral element method (SEM) or the discontinuous Galerkin finite element method (DGFEM). The frequency and angle depen- dent boundary impedance is mapped to a multipole model and formulated in differential form. The solution of the boundary differential equations comes with minimal computational cost. In the ER model, wave splitting is applied at the boundary to separate the incident and reflected parts of the sound field. The directional properties of the incident sound field are determined from the incident particle velocity and the boundary conditions are adjusted continuously according to the wave angle of incidence. The accuracy of the boundary condition model is assessed by comparing simulations against measurements, where a significantly improved match between simulations and measurements is found when the ER model is used.
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Discrete Transparent Boundary Conditions for Systems of Evolution Equations

Discrete Transparent Boundary Conditions for Systems of Evolution Equations

This dissertation is concerned with the derivation and implementation of discrete trans- parent boundary conditions for systems of evolution equations. Transparent boundary con- ditions (TBCs) are a special kind of artificial boundary conditions, that are constructed in such a way, that the solution on a bounded domain with TBCs is equal to the solution of the whole-space problem restricted to the bounded (computational) domain. The partial differential equations are discretised by finite differences (θ-scheme) and discrete transpar- ent boundary conditions (DTBCs) are constructed for the discrete equation. Therefore, the DTBCs are well adapted to the numerical scheme. For scalar equations these DTBCs are well established. Compared to discretising the analytical TBC, in the scalar case it is known that these DTBCs have the advantage, not to destroy the stability properties of the underlying discrete scheme and to avoid any numerical reflections. In this dissertation we will deal with systems of partial differential equations (parabolic and Schr¨odinger type). For these systems the approach of DTBCs is completely new and involves additional prob- lems not encountered in the scalar case. Since the numerical computation of these DTBCs is very costly, we give an approximation which greatly reduces the effort.
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Failure of the $N$-wave interaction approximation without imposing periodic boundary conditions

Failure of the $N$-wave interaction approximation without imposing periodic boundary conditions

proving the failure of the NLS approximation for the water wave problem with suitably chosen small surface tension and periodic boundary conditions. The proof given in [SSZ15] is unsatisfactory in the sense that an approximation theorem beyond the natural time scale for an extended TWI system has to be established and the qualitative behavior of this high-dimensional amplitude system has to be discussed.

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Adaptive boundary conditions for exterior stationary flows in three dimensions

Adaptive boundary conditions for exterior stationary flows in three dimensions

density and the viscosity , are arbitrary positive constants. From , and u 1 we can form the length ` = =( u 1 ), the so called viscous length of the problem. The viscous forces and the inertial forces are quantities of comparable size if the Reynolds number Re = A=` is neither too small nor very large. Below, when solving the problem (1) numerically for the example case where ~ B is a prism, we restrict the equations (1) from the exterior in…nite domain ~ to a sequence of bounded domains ~ D ~ and study the precision of the results as a function of the domain size, once with naïve boundary conditions on the surface ~= @ ~ D n @ ~ B of the truncated domain and once with the newly proposed adaptive boundary conditions. Note that, in contrast to the …nite volume case, the boundary conditions at in…nity do not prescribe the total ‡ux of ‡uid (from left to right say). In particular, it does not follow from lim jxj!1 (~ u(x) ~ u 1 ) = 0, that lim
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Adding boundary conditions on the bed for Telemac-3D simulations

Adding boundary conditions on the bed for Telemac-3D simulations

to the source terms, and once it is calculated the velocity on the bed is fixed to zero, as source terms do not cancel the impermeability of the bed. In essence, the simulation launched by file bottom_inlet_equiv_source should give the same results as t3d_bottom_source.cas and it is used to validate the fact that no bug has been introduced in the development of the liquid bed boundary conditions.

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POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions

POD-Galerkin Modeling for Incompressible Flows with Stochastic Boundary Conditions

The boundary treatment presented in Gunzburger et al. (2007) is general- ized in order to be applicable to problems with parametrized stochastic Dirich- let boundary conditions. The method of Gunzburger et al. recombines the POD basis functions so that homogeneous Dirichlet conditions are enforced at selected boundary points. This leads to modified POD basis functions which fulfill homogeneous Dirichlet conditions under the following conditions: Firstly, the Dirichlet boundary must consist of non-overlapping segments. Secondly, the Dirichlet data on each segment must be given by a spatial func- tions times a possible time-dependent parameter. Additionally, the original method requires the selection of one boundary point per segment, which cor- responds to inhomogeneous Dirichlet data unless homogeneous conditions are prescribed at the complete segment. The generalized method builds on the idea that homogeneous Dirichlet conditions can be enforced directly on the union of all Dirichlet boundaries. This leads to a relaxation of the necessary conditions. In particular, the Dirichlet data on the union of all boundary seg- ments must be expressible as a linear combination of spatial functions with possibly time-dependent coefficients. This exactly corresponds to the way in which random fields are presented after a Karhunen-Loève expansion has been applied. Moreover, in the new method it is not necessary anymore to select certain points at which homogeneous conditions are enforced.
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Non-Reflecting Boundary Conditions for Traffic Flow Schrödinger Equation

Non-Reflecting Boundary Conditions for Traffic Flow Schrödinger Equation

0.1. The discretisation steps have been chosen as follows: dx = 0.1, dt = 0.001. The temporal evolution of this function has been studied with and without the non-reflecting boundary described above. The result in Fig. 1 demonstrates a nice correspondence between the non-reflecting boundary conditions at x = 0 (green line) and the situation without boundary (red line).

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Engine Inflow Boundary Conditions for Specification of Mass Flow

Engine Inflow Boundary Conditions for Specification of Mass Flow

Having now complete information about the characteristic speeds and variables, it is possible to derive permeable boundary conditions as follows: by considering the signs of the characteristic speeds, determine how many flow quantities are transported onto the boundary, and how many are transported away. For example in the case of an engine inflow the flow is subsonic in the direction of the boundary, hence λ + and λ 0 are positive while λ − is negative, implying that Λ + and Λ 0 are transported onto the boundary from the field, and Λ − is determined at the boundary. Hence one may choose a single flow variable to be set on the boundary termed the physical boundary condition; in general one must choose a set of variables corresponding to the number of negative eigenvalues 1 .
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Methodology for holistic evaluation of building energy systems unter dynamic boundary conditions

Methodology for holistic evaluation of building energy systems unter dynamic boundary conditions

Building Energy Systems under Dynamic Boundary Conditions This dissertation proposes an extended evaluation methodology with respect to CO 2 -emissions for the holistic and time-resolved assessment of energy systems at building level that considers the different system levels, their coupling and technical restrictions. In this context, the growing integration of renewable energy sources and the introduction of novel, efficient technologies for heating in the building sector will cause fundamental changes in the operation, control and evaluation of the energy system. The operational flexibility of building energy systems by thermal and electrical storage is included in the presented evaluation method to enable a comparative assessment of the system´s performance. Results demonstrate, that a grid-supporting operation mode of building-energy systems is possible, enabling an enhanced integration of renewable energy sources inside the grid and consequently influencing the allocation of CO 2 -emissions. Additionally, the results indicate, that grid-supportive measures of building energy systems have not been sufficiently considered in former evaluation methodologies. The presented evaluation method is embedded in a flexible simulation environment to study different scenarios, taking a step forward to a possible integration in evaluation standards and regulations.
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Gutenberg Open Science: Thermal fluctuations and boundary conditions in the lattice Boltzmann method

Gutenberg Open Science: Thermal fluctuations and boundary conditions in the lattice Boltzmann method

Apart from the diffusive reflections, many other attempts to improve the accuracy of the bounce-back boundary condition have been proposed. One class of these approaches tries to find a solution for the unknown populations in terms of the populations on adjacent lattice sites. Let us refer to this class as closure schemes. Their aim is to generate a set of pop- ulations that satisfies the desired boundary conditions at the hydrodynamic level, i.e., the desired velocity field (Dirichlet condition) and its gradients (Neumann condition). Ziegler [132] combined the nodal bounce-back with setting the grazing directions to the average of the incoming directions. This scheme ensures the no-slip condition by construction, but it is not mass conserving on the boundary nodes. Skordos [133] addressed the problem of in- versely mapping the hydrodynamic fields to the lattice Boltzmann populations. A modified collision operator was introduced for the boundary nodes, which relaxes the populations towards an equilibrium distribution that includes velocity gradients as additional correction terms. Although a modified equilibrium distribution at the boundary is a reasonable as- sumption, the inclusion of gradient terms is questionable and lacks a rigorous justification in terms of the Chapman-Enskog expansion. If the velocity gradients are unknown, they must be evaluated using finite-differences. Moreover, the density was assumed to be known at the boundary nodes, which may not always be appropriate. Noble and coworkers [134, 135] developed a two-dimensional closure scheme where the density is a computed quantity and only the velocity components at the boundary have to be prescribed. The scheme is based on dividing the populations into groups that stream in from neighboring fluid nodes, boundary nodes or solid nodes, respectively. The latter of these three are the unknown quantities in an equation system which is obtained from the conservation laws for mass and momentum. Noble et al
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Regularity of stationary surfaces of Cartan functionals with Plateau boundary conditions

Regularity of stationary surfaces of Cartan functionals with Plateau boundary conditions

However, the next step of proving global H¨ older continuity is not trivial at all. In [DHKW92b] for the area functional and in [HvdM03b] for Cartan functionals, this result was achieved by a comparison with a (locally) harmonic surface possessing the same boundary values as X on a small ball. We are not able to compare to other surfaces without the minimization property, but have to stick to the weak Euler-Lagrange inequality. Going this way, we have to preserve the Plateau boundary conditions after a pertubation of the form X + tϕ t , especially

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Dynamics of stochastic partial differential equations with dynamical boundary conditions

Dynamics of stochastic partial differential equations with dynamical boundary conditions

This section deals with the most general assumptions on the noise. Instead of simple additive noise, we have more general multiplicative noise. In fact, we need that the drift term has to fullfill some trace condition, see Hypothesis 8.5.6[vii]. Otherwise, it is not possible to use integration by parts successfully. There are two different approaches included in this section. Both the approach of mild and weak solutions is considered. Weak solutions were analyzed in the work by Keller [33]. His ansatz is generalized to dynamical boundary conditions in this chapter. Following the theory of weak solutions, we can show the existence of a random attractor on (8.12). Additionally, there exists to each lemma considering weak solutions an associated remark, which proves the statement of the corresponding lemma, if the reader only considers mild solutions.
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The Immersed Interface Technique for Parabolic Problems with Mixed Boundary Conditions

The Immersed Interface Technique for Parabolic Problems with Mixed Boundary Conditions

In this work, we use an immersed interface technique to derive an implicit finite difference scheme in space and time for a parabolic problem with mixed boundary conditions. The space discretization is motivated by the scheme developed for the analogous problem in the elliptic context (see [2]), taking into account that the Lapla- cian of the solution is not available for parabolic problems. A proof of convergence is given, based on a maximum principle satisfied by the discrete operator. Since there are some positive off-diagonal entries in the matrix associated with the discrete oper- ator, this matrix is not an L-matrix but a perturbation of an M -matrix. A technique similar to the one described in [1] is then used to show the monotonicity of the sys- tem. This step induces the condition δt > Ch 2 . This condition is much less restrictive than the conventional conditions between the time-step and the mesh size (such as the CFL condition in the hyperbolic context, for example). Note that this condition is just opposite to δt  h 2 which is used to show the convergence for explicit schemes in the parabolic context.
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Experimental modal analysis of aeroelastic tailored rotor blades in different boundary conditions

Experimental modal analysis of aeroelastic tailored rotor blades in different boundary conditions

Abstract. Bend-twist coupled blades are intended to reduce the loads on the overall wind turbine by passively adapting to current wind conditions. The coupling results from complex- design shapes and structures using advanced finite element models utilising shell and volume elements. These models are however prone to mispredict the structural dynamic behaviour of the rotor blades. In particular, normal modes with both bending and torsion contributions, as well as local vibrations of the blade shells include computational uncertainties. Therefore, in order to update flawed model parameter assumptions, a modal characterisation of blade prototypes including mode shapes is essential. In the present study results of a modal test campaign involving four identical rotor blades of 20 m length with geometric bend-twist coupling are reported. Design, realisation, and post-processing of the experiments have been carried out under careful consideration of a pre-existing FE shell model. Modal data is obtained at two different stages of the manufacturing process and for one blade in two separate boundary conditions, i.e. free-free in elastic suspensions and clamped to a test rig. Due to the high sensor density in both configurations, the identified normal modes do not only include coupled eigenforms but also mode shapes illustrating cross-sectional vibrations; the latter attributed to the deflection of the blade shells. The acquired dataset is found to be well-suited for the validation of the numerical model and represents a reliable basis for updating.
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Maximal Regularity in Weighted Spaces, Nonlinear Boundary Conditions, and Global Attractors

Maximal Regularity in Weighted Spaces, Nonlinear Boundary Conditions, and Global Attractors

concentration waves from Γ into an infinitesimal layer near the boundary. Similar dynami- cal boundary conditions arise in Cahn-Hilliard or Caginalp phase field models if one takes into account the short-ranged interaction with walls [73]. They also arise in two phase flows with soluble surfactant [14]. In the literature these boundary conditions are also called gen- eralized Wentzell boundary conditions [43]. Semilinear versions of (5) with a single equation were investigated by many researchers, for instance by Favini, J. A. Goldstein, G. R. Gold- stein & Romanelli [38, 39, 40], Sprekels & Wu [80] and Vazquez & Vitillaro [83]. Results on quasilinear versions do not seem to exist. There are further results on quasilinear systems with dynamic boundary conditions of reactive type, i.e., where tangential derivatives do not occur. A dynamic theory for such problems was established by Escher [36], based on Amann’s work. We refer to Constantin & Escher [20] and the references therein for more recent developments.
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Absorbing numerical boundary conditions for the Dirac equation / Markus Fleck

Absorbing numerical boundary conditions for the Dirac equation / Markus Fleck

thesis to successfully apply this scheme of boundary conditions (which do not need to process the history of the wave function) to the Dirac equation. Furthermore, the stability and well absorbing quality of the boundary conditions is studied by simulations investigating Klein’s paradox. Klein’s paradox is a feature of relativistic wave equations and is discussed in greater detail. The well-behaving of the boundary conditions in simulations of both pure and mixed states (density matrices) is shown. One simulation involving a step potential demonstrates the possibility to apply a time-dependent voltage across the outside regions using these boundary conditions.
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Discrete artificial boundary conditions

Discrete artificial boundary conditions

In the literature the discrete approach did not gain much attention yet. The first discrete derivation of artificial boundary conditions was presented in [D2, Section 5]. This discrete ap- proach was also used in [D5], [D6], [D7] for linear hyperbolic systems and in [D3] for the wave equation in one dimension, also with error estimates for the reflected part. In [D6] a discrete (nonlocal) solution operator for general difference schemes (strictly hyperbolic systems, with constant coefficients in 1D) is constructed. Lill generalized in [D4] the approach of Engquist and Majda [D2] to boundary conditions for a convection–diffusion equation and drops the stan- dard assumption that the initial data is compactly supported inside the computational domain. However, the derived –transformed boundary conditions were approximated in order to get local–in–time artificial boundary conditions after the inverse –transformation.
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Periodic Boundary Conditions and the Error-Controlled Fast Multipole Method

Periodic Boundary Conditions and the Error-Controlled Fast Multipole Method

Fast multipole methods have been available since 1987. However, the FMM rarely earned huge appreciation in the scientific community. Many scientific articles claimed that the advantage of the linear complexity will only be visible for very large systems with millions or even billions of particles [62]. These authors stated that large prefactors and the schemes to obtain the FMM parameter set will slow down calculations for a moderate number of particles. Since these statements strongly depend on the implementation and not on the FMM theory itself, in this chapter we will shed light on this claim and compare our error-controlled implementation against freely available codes from other groups dealing with long-range interactions. Thereby, we will show that the expected theoretical linear scaling along with high precision calculations are feasible – for open and mixed periodic boundary conditions as well.
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Fluidics of thin polymer films : boundary conditions and interfacial phenomena

Fluidics of thin polymer films : boundary conditions and interfacial phenomena

Author contributions: K. Jacobs designed research. Experiments were performed by O. B¨aum- chen (experiments on the AF 1600 substrate) and R. Fetzer (experiments on OTS and DTS). The article was written by O. B¨aumchen. Writing was supervised by R. Fetzer and K. Jacobs. Abstract - The control of hydrodynamic boundary conditions has become more and more important for confined geometries such as lab-on-a-chip devices. Probing the boundary condi- tions at the solid/liquid interface is therefore of essential interest. In this article we study the dewetting dynamics of thin polymer film flow on smooth hydrophobic surfaces and present a method to extract hydrodynamic boundary conditions, i.e. the slip length. It has been shown that different energy dissipation mechanisms occur in such systems. Viscous dissipation and friction at the solid/liquid interface counteract the capillary driving force. Silicon wafers with three different hydrophobic surface coatings, an amorphous Teflon r coating (AF 1600) and self-assembled monolayers of octadecyltrichlorosilane (OTS) and dodecyltrichlorosilane (DTS) exemplarily show the applicability of the proposed model. By superposing both dissipation mechanisms and plotting hole growth dynamics data in a special way, we are able to identify viscous part and slippage part. Thereby we manage to extract the slip length for our systems. Concerning the silane layer surfaces we find values in the range of microns that decrease with increasing temperature. Especially on DTS slippage is pronounced; the slip lengths are about one order of magnitude larger than on OTS. In case of the AF 1600 coating, viscous dissipa- tion dominates and we obtain slip lengths roughly between 10 to 100 nm. Additionally, further potential dissipation mechanisms will be briefly discussed.
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Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains

Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains

Among all possible boundary conditions, Dirichlet boundary conditions are the most popular, meaning that one prescribes values for u on the boundary of Ω. Regularity of solutions of Dirichlet problems lies at the very heart of potential theory, and sharp conditions are known under which the solution is continuous up to the boundary of Ω. For the linear case, this is the celebrated Wiener criterion. Quasilinear generalizations have been found and studied by Maz’ya [Maz70], Gariepy and Ziemer [GZ77], and Kilpeläinen and Malý [KM94]. In fact, much is known about the regularity of the solution and its derivatives even if the right hand side is very rough, see for example recent articles by Mingione [Min07, Min10] and Duzaar and Mingione [DM09]. For Robin boundary conditions, on the other hand, and even for the special case of Neumann boundary conditions, i.e., if h(u) = 0 in (1.3), the situation is not as well understood. There are, however, results due to Lieberman [Lie83, Lie92] if the domain is smooth except for a small set. One of the main goals of this thesis is to establish regularity up to the boundary also for these boundary conditions if Ω is a Lipschitz domain. More precisely, we want to show that every solution is Hölder continuous up to the boundary of Ω. This means that u allows for a continuous extension to Ω which is Hölder continuous for the same exponent and the same Hölder constant.
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