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Architectural Acoustics
Published in Malcolm J. Crocker, A. John Price, Noise and Noise Control, 2018
Malcolm J. Crocker, A. John Price
If the reflecting surface is not a plane, but curved, then the rays will be focused in a similar manner to that found in optics for reflection by curved mirrors. In the case of a concave reflecting surface, sound rays will be focused to a point forming a real acoustic image. This can be most undesirable acoustically, since it will result in a "hot spot" within the room which will be very loud while it would be considerably quieter at other points away from the focus in the room. Figure 5.4 shows that the image produced by a concave reflector can be located by using the fact that rays traveling parallel to the axis will be reflected through the focus (equal to one half of the radius of curvature R) and conversely, rays travelling through the focus will be reflected parallel to the axis. Again, in order for this focusing effect to take place the dimensions of the concave surface must be larger than a wavelength (see Equation 1.27) and the surface must be smooth and reflective. This effect must be avoided in architectural acoustics since it leads to hot spots and echoes. Many auditoria have concave rear walls or balconies with concave front walls which would give rise to echoes at the front of the audience or even on the stage if not properly treated to be made highly absorbent. Domes and shell ceilings can cause similar effects and almost always result in an acoustic problem.
Environmental Design
Published in Shatha N. Samman, Human Factors and Ergonomics for the Gulf Cooperation Council, 2018
While there is no single acoustic solution that can be universally applied, spatial planning, room shaping, and architectural materials are fundamental elements in the design of acoustic performance. Since sound moves through building spaces in a variety of ways, the shape of rooms and indoor materials used may be selected and assembled to control sound and influence reverberation time (i.e., the way a space “sounds”). Specifically, concave surfaces tend to concentrate or focus reflected sound in one area, whereas convex surfaces tend to disperse sound in multiple directions. Additionally, the use of absorptive and diffusing materials can control the amount of reflected sound within a room, diminishing the strength of the sound and diffusing its effects. In small square rooms, for example, low frequencies seem to be predominant, so that materials placed on the ceiling surface may lessen any annoying speech effect; in round rooms, sound-diffusing elements should be positioned on curved surfaces to mitigate and disperse sound in many directions (GSA, 2011; Knauf, 2013). Furthermore, panels, doors, and partitions can be used as physical barriers to block sound between spaces. Electronic background sound may be introduced to mask sounds from air conditioning systems and other equipment, or indirect speech within a space, so that it is either not intelligible or even audible to unintended listeners (ASA, 2016). See Table 7.1 for the influence of individual differences (i.e., age, cultural background) on sound comfort.
On the convexity of yield and potential surfaces in rotational hardening critical state models
Published in António S. Cardoso, José L. Borges, Pedro A. Costa, António T. Gomes, José C. Marques, Castorina S. Vieira, Numerical Methods in Geotechnical Engineering IX, 2018
Jon A. Rønningen, Grimstad Gustav, Nordal Steinar
In this expression, M is directly a function of the modified Lode angle, while α might change due to kinematic hardening. It can be seen in the equation above that whenever α approaches M (i.e. due to kinematic hardening) the expression g(θα,α) becomes very large and approaches infinity as a limit. As stated in (Crouch and Wolf, 1995) the introduction of a dependency of the anisotropic bounding surface on the Lode angle is not straightforward. In the work on this subject it is discovered that due to the specific form of eq. (4) and (1), the shape of the surface could actually become concave as it is “sheared” (Figure 3 and Figure 4). This may happen even if the shape is convex for α equal to zero. In general, concave surfaces are undesirable in plasticity theory and could lead to numerical problems.
Double-tolerance design for manufacturing systems
Published in IISE Transactions, 2021
Di Liu, Tugce Isik, B. Rae Cho
A numerical example for the optimization problem and its sensitivity analysis are shown in Section 6.1. Here, we present an example of the objective function y with fixed parameter values. Figure 5 shows that when is less than an inflection point, the objective function is concave in and it is convex in when is greater than the inflection point. However, one can compute the limit of the objective function y as and tend to infinity so that the products are neither reworked nor scrapped, which provides the net profit of the system when the long-run average production costs are at a lower bound:
Utility maximisation for resource allocation of migrating enterprise applications into the cloud
Published in Enterprise Information Systems, 2021
Here, provider-capacity constraints in the PP is removed by subtracting the objective to the penalty function . The penalty function would be close to zero as and increase to infinity as . With the penalty function, the relaxation optimisation problem (5) is indeed equivalent to the PP with explicit linear provider-capacity constraints. The price function is supposed to be increasing and continuous. Then, is a convex function, and is concave because of concavity of .
Fenchel–Rockafellar theorem in infinite dimensions via generalized relative interiors
Published in Optimization, 2023
D. V. Cuong, B. S. Mordukhovich, N. M. Nam, G. Sandine
Let be a convex function and let be a concave function. The convex Fenchel conjugate of f is the function given by The concave Fenchel conjugate of g is the function given by