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.

Mehr anzeigen
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.

Mehr anzeigen
131 Mehr lesen

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.

27 Mehr lesen

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

Mehr anzeigen
20 Mehr lesen

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**.

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.

Mehr anzeigen
197 Mehr lesen

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).

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 .

Mehr anzeigen
26 Mehr lesen

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.

Mehr anzeigen
202 Mehr lesen

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

Mehr anzeigen
195 Mehr lesen

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

198 Mehr lesen

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.

Mehr anzeigen
145 Mehr lesen

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.

Mehr anzeigen
20 Mehr lesen

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.

Mehr anzeigen
12 Mehr lesen

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.

Mehr anzeigen
221 Mehr lesen

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**.

Mehr anzeigen
98 Mehr lesen

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.

Mehr anzeigen
125 Mehr lesen

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.

Mehr anzeigen
136 Mehr lesen

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.

Mehr anzeigen
161 Mehr lesen

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.

Mehr anzeigen
130 Mehr lesen