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Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
Let {Xi:i∈I} $ \{X_i: i \in \mathfrak I \} $ be a family of topological spaces and set X:=∏i∈IXi. $ { X:= \prod \nolimits _{i\in \mathfrak I } X_i. } $ The product topology on X is the initial topology with respect to the family of projection mappings πi:X→Xi $ \pi _i:X \rightarrow X_i $ . By 0.6.4(c), a net (fα) $ (f_\alpha ) $ in X converges to f in this topology iff fα(i)=πi(fα)→πi(f)=f(i) $ f_\alpha (i) = \pi _i(f_\alpha ) \rightarrow \pi _i(f) = f(i) $ for each i∈I $ i\in \mathfrak I $ . For this reason the product topology is also called the topology of pointwise convergence on I $ \mathfrak I $ . Note that the product topology on Rd $ \mathbb R ^{d} $ is simply the topology defined by the Euclidean metric.
Design of a distributed compliant mechanism using spring-lever model and topology optimization for piezoelectrically actuated flapping wings
Published in Mechanics of Advanced Materials and Structures, 2021
Nilanjan Chattaraj, G. K. Ananthasuresh, Ranjan Ganguli
In the previous section, the topology optimization yields an initial topology of the flapping mechanism for the given specification, which is obtained by the SL model discussed in the preceding section. Now we have to fine-tune this initial topology depending on the sensitivity of material distribution across the branches. This process eliminates few branches of the initial topology for having low distribution of material, and thereby yields the final topology. Now, the final topology is transferred to an actual 3D design using a CAD software (Comsol Multiphysics®) to analyze the performance of the mechanism in detail. We have also recreated the final topology using a beam model in Matlab and have examined the design considering the geometric nonlinearity of the beam [39] and [40]. Finally a nonlinear finite element model, which is based on the total Lagrangian approach, is used in Matlab® platform to examine the performance of the final mechanism considering the geometric nonlinearity of the beam model.
Mesh modeling and simulation for three-dimensional warp-knitted tubular fabrics
Published in The Journal of The Textile Institute, 2022
Haisang Liu, Gaoming Jiang, Zhijia Dong
The circumference is naturally divided into four parts clock-wise by the coordinate x-z. It is necessary to make a brief classified discussion on rotation angles owing to the differential rotation direction of stitch topology in separate areas. In parts I and IV, the final topology is obtained by the rotation counterclockwise and on the contrary, the initial topology rotates clockwise. The rotation angle is distinguished by the position part of Qn, which is signified as follows：