The Implementation of High-Order Single-Step Integration Technique into Circuit Simulator

Author(s):  
M. M. Gourary ◽  
S.G. Rusakov ◽  
S.L. Ulyanov ◽  
M. M. Zharov
Polymers ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 3041
Author(s):  
Ji Hwan Kim ◽  
Joung Sook Hong ◽  
Akira Ishigami ◽  
Takashi Kurose ◽  
Hiroshi Ito ◽  
...  

A ternary composite of poly(lactic acid) (PLA), poly(caprolactone) (PCL), and carbon black (CB) shows the PCL-induced CB self-aggregation and percolation formation when the amount of the PCL phase as the secondary phase is as small as the amount of CB. Furthermore, when the drop size of the PCL phase becomes smaller, the ternary composite forms a percolation of high order structure, resulting in a remarkable enhancement of the electrical conductivity (~4 × 10−2 S/m with 4 wt.% CB). To further control the percolation structure, the composite fabrication is controlled by splitting a typical single-step mixing process into two steps, focusing on the dispersion of the secondary PCL phase and the CB particles separately. Under the single-step mixing protocol, the ternary composite shows a structure with greater CB aggregation in the form of a high aspect ratio and large aggregates (aggregate perimeter~aggregate size 0.7). Meanwhile, the two-step mixing process causes the CB aggregates to expand and create a higher structure (aggregate perimeter~aggregate size 0.8). The reduced size of the secondary phase under a mixing condition with high shear force prior to the addition of CB provides a larger interfacial area for CB to diffuse into the PCL phase during the subsequent mixing step, resulting in a further expansion of CB aggregation throughout the composite. The particle percolation of such a high order structure is attributed to high storage modulus (G′), high Young’s modulus, high dielectric loss (ε″), and negative–positive switching of dielectric constant at high frequency (of 103 Hz) of composite.


2007 ◽  
Vol 1 (5) ◽  
pp. 1024 ◽  
Author(s):  
F.P. Hart ◽  
S.R. Luniya ◽  
J. Nath ◽  
A. Victor ◽  
A. Walker ◽  
...  

Author(s):  
Lonny L. Thompson ◽  
Prapot Kunthong

The time-discontinuous Galerkin (TDG) method possesses high-order accuracy and desirable C-and L-stability for second-order hyperbolic systems including structural acoustics. C- and L-stability provide asymptotic annihilation of high frequency response due to spurious resolution of small scales. These non-physical responses are due to limitations in spatial discretization level for large-complex systems. In order to retain the high-order accuracy of the parent TDG method for high temporal approximation orders within an efficient multi-pass iterative solution algorithm which maintains stability, generalized gradients of residuals of the equations of motion expressed in state-space form are added to the TDG variational formulation. The resultant algorithm is shown to belong to a family of Pade approximations for the exponential solution to the spatially discrete hyperbolic equation system. The final form of the algorithm uses only a few iteration passes to reach the order of accuracy of the parent solution. Analysis of the multi-pass algorithm shows that the first iteration pass belongs to the family of (p+1)-stage stiff accurate Singly-Diagonal-Implicit-Runge-Kutta (SDIRK) method. The methods developed can be viewed as a generalization to the SDIRK method, retaining the desirable features of efficiency and stability, now extended to high-order accuracy. An example of a transient solution to the scalar wave equation demonstrates the efficiency and accuracy of the multi-pass algorithms over standard second-order accurate single-step/single-solve (SS/SS) methods.


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