Asymmetric Structure Research Articles

Investigating Static and Dynamic Behaviors in 3D Chiral Mechanical Metamaterials by Disentangled Generative Models

AbstractChiral mechanical metamaterials with engineered asymmetric structures have garnered significant interest owing to their potential for manipulating mechanical waves and unconventional mechanical properties. Although several design methodologies have made significant strides in the design and discovery of chiral mechanical metamaterials, 3D designs that incorporate multiple mechanical characteristics remain largely unexplored and the static and dynamic properties of these structures have not received sufficient attention. Here, an innovative approach is introduced for the inverse design of 3D chiral mechanical metamaterials with desired static and dynamic properties, leveraging streamlined shapes to enable a versatile design. By employing conditional generative adversarial networks (c‐GANs), the desired mechanical properties are targeted by focusing on both wave attenuation and stress distribution under compression and shear loading. 3D chiral mechanical metamaterials with additive manufacturing are fabricated to demonstrate the performance of the proposed c‐GAN. Experimental investigations show that the generated structures exhibited a wide range of wave attenuation properties, low stress concentrations, and a variety of stress‐strain curve profiles including nonlinear stiffness, demonstrating their effectiveness in achieving the desired mechanical characteristics. This research offers significant advancements in the on‐demand design capabilities of metamaterials and their applications.

Patterned Ultraslippery Surfaces of Stainless Steel Prepared by Femtosecond Laser Ablation for Directional Manipulation of Liquid Droplets.

Slippery liquid-infused porous surfaces (SLIPS) have promising applications in chip laboratories, nanofriction power generation, and microfluidics due to their excellent properties such as good hydrophobicity and low adhesion. However, the self-driven stability of conventionally lubricated surfaces is not high, and the velocity of liquid droplets is difficult to regulate. This greatly limits the potential applications of SLIPS. A strategy is offered to prepare microporous structures of SLIPS directly on a stainless-steel substrate using femtosecond laser processing technology as the main means to realize exhibiting smoothness to liquids. At the same time, the principle of bionics is utilized, the porous structure of SLIPS is combined with the groove structure of rice leaves, or porous structures are combined with the wedge structure of shorebird beak to prepare the three-dimensional structure of SLIPS. Droplets exhibit significant individual anisotropy on three-dimensional (3D) SLIPS of leaf-like groove stripe structure in rice, enabling the precise control of droplet motion direction. When droplets are transported in wedge-shaped SLIPS with an asymmetric structure, the wedge edge can limit the direction of droplet motion while squeezing the droplet to generate Laplace pressure gradient, which achieves continuous self-driven transport of droplets. In addition, based on the above two processing strategies, an information transfer device is designed: the splicing of the self-driven transport surface with anisotropic topological channels enables the differential drive for liquid transport, which provides the conditions for the information transfer of the droplets. This strategy not only is simple and efficient but also provides new ideas for the effective development of multifunctional SLIPS as well as lab-on-a-chip and microfluidic domains.

Asymmetric structural strategy enables efficient MR-TADF emitters with low efficiency roll-off and doping-concentration insensitivity

Multiple resonance (MR) thermally activated delayed fluorescence (TADF) materials containing B/N hold great promise for application in efficient narrowband organic light-emitting diodes (OLEDs). However, overcoming the challenges of reducing efficiency roll-off while minimizing aggregation-caused quenching to achieve highly efficient, low efficiency roll-off and doping concentration-insensitive MR-TADF emitters remains a formidable task. To this end, we designed two asymmetric B/N MR-TADF emitters by integrating a twisted heptagonal tribenzo[b,d,f]azepine (TBA) unit with a bulky substituted carbazole unit. The new compounds inherit the excellent properties of both parent skeletons. BN-TPCzTBA achieves a high photoluminescence quantum yield of up to 98.6 % and attains a maximum external quantum efficiency (EQE) of up to 36.8 % without the use of a sensitizer. The EQE remains at 22.8 % at a high brightness of 1000 cd m−2, demonstrating a low efficiency roll-off (37 %). Importantly, the twisted, bulky, and asymmetrical structure of these molecules allows them to sustain over 30 % of the EQEmax with a consistent full width at half maximum (FWHM) even at doping concentrations of up to 30 %. This thereby presents an innovative molecular design strategy that addresses critical issues in MR-TADF material engineering, combining low efficiency roll-off with doping concentration insensitivity.

Hydrogen Bond‐Induced Flexible and Twisted Self‐Assembly of Functionalized Carbon Dots with Customized‐Color Circularly Polarized Luminescence

Circularly polarized luminescence (CPL) is widely applied in optical data storage, quantum computing and backlights in three‐dimensional (3D) displays. Carbon dots (CDs) exhibit competitive optical properties, in addition to excellent resistance to photo‐ and chemical‐bleaching after carbonization. Combining the superior optical performance with polarization peculiarities through hierarchical structure engineering is imperative for the development of CDs. Here, oriented assembly was driven by hydrophobic interactions of aromatic ligands, which participated in the surface‐ligand post‐modification process on ground‐state chiral carbon core. Furthermore, the residual chiral amides on CDs formed multi‐hydrogen bonds during gradual aggregation, causing the assembled materials to form asymmetric bending structure. Superficial ligands interfered with optical dynamics of exciton radiation transition and promoted the excited state of the assembled materials to achieve a circularly polarized signal. The linkage ligands successfully overcame the frequent phenomenon of aggregation‐induced quenching and contributed further to the formation of self‐supporting films by assembly and facilitated chiral optical expression. The full‐color and white CPL were manipulated by simply regulating the functional groups on the ligands. Finally, based on the stable chiral powder phosphors, large chiral flexible films and multicolor chiral light‐emitting diodes were constructed which provide feasible materials and technical support for flexible 3D displays.

Design and performance evaluation of toroidal TLCDs in bidirectional seismic control of structures using a modified dynamic model

In this study, a modified dynamic model which distinguishes between liquid oscillation and sloshing in vertical columns is derived for toroidal tuned liquid column dampers (TLCDs). In the modified model of toroidal TLCDs, the primary sloshing mode within vertical columns is modeled as mass–spring-dashpot systems, whereas the oscillatory behavior of the liquid is analyzed employing the conventional theory of TLCDs. The accuracy of the modified dynamic model in capturing bidirectional liquid responses is verified by computational fluid dynamics (CFD)-based simulations. A large number of CFD-based simulations are performed to establish a ready-to-use suggested formula for a newly introduced parameter (which is related to the geometric configuration of liquid containers and the initial liquid depth) in the modified model. Subsequently, considering the maximum allowable liquid displacement in vertical columns, an optimized design approach for toroidal TLCDs installed in symmetric and asymmetric structures is illustrated. On the basis of the proposed optimization design method, the bidirectional vibration control effects of toroidal TLCDs are comprehensively analyzed under El Centro, Loma Prieta, Northridge, and Chi–Chi seismic excitations in both the time and frequency domains. The results demonstrate the effectiveness of toroidal TLCDs for bidirectional seismic control of structures.

Accretion versus core-filament collision. Implications for streamer formation in Per-emb-2

Recent millimetre and sub-millimetre observations have unveiled elongated and asymmetric structures around protostars. These structures, referred to as streamers, often exhibit coherent velocity gradients, seemingly indicating a directed gas flow towards the protostars. However, their origin and role in star formation remain uncertain. The protostellar core Per-emb-2, located in Barnard 1, has a relatively large streamer of $10^4$ au that is more prominent in emission from carbon-chain molecules. We aim to unveil the formation mechanism of this streamer. We conducted mapping observations towards Per-emb-2 using the Nobeyama 45 m telescope. We targeted carbon-chain molecular lines such as CCS, HC$_3$N, and HC$_5$N at 45 GHz. Using astrodendro we identified one protostellar and four starless cores, including three new detections, on the Herschel column density map. The starless and protostellar cores are more or less gravitationally bound. We discovered strong CCS and HC$_3$N emissions extending from the north to the south, appearing to bridge the gap between the protostellar core and the starless core to its north. This bridge spans $3 10^4$ au with velocities of 6.5 to 7.0 km $. The velocity gradient of the bridge is opposite that of the streamer. Thus, the streamer is unlikely to be connected to the bridge, suggesting that the former does not have an accretion origin. We propose that a collision between a spherical core and the filament has shaped the density structure in this region, consequently triggering star formation within the head-tail-shaped core. In this core-filament collision scenario, the collision appears to have fragmented the filament into two structures. The streamer is a bow structure, while the bridge is a remnant of the shock-compressed filament. Thus, we conclude that the Per-emb-2 streamer does not significantly contribute to the mass accumulation towards the protostar.

Polarization-independent quasi-BIC supported by non-rotationally symmetric dimer metasurfaces.

Asymmetric metasurfaces supporting quasi-bound states in the continuum (-BICs) have recently attracted significant interest in the field of nanophotonics due to their high quality factor and strong light-matter interaction properties. However, asymmetric metasurface structures are susceptible to the polarization state of the incident light, which constrains their potential applications. In this Letter, we present a new, to our knowledge, scheme of polarization-independent quasi-BIC resonance supported by a non-rotationally symmetric nanorod dimer metasurface. By tuning the asymmetry parameter, the designed metasurface exhibits a consistent quasi-BIC response for incident plane waves of arbitrary polarization. The physical mechanism of the quasi-BIC resonance is elucidated by the study of the far-field multipole decomposition and the near-field electromagnetic distribution. We then point out that the realization of the polarization-independent quasi-BIC resonance depends on the transition between magnetic and electric quadrupoles. Furthermore, the designed metasurface is demonstrated to have excellent refractive index sensing performance. This work provides a new idea for the design of polarization-independent and high-performance resonators.

Templated trimerization of the phage L decoration protein on capsids.

The 134-residue phage L decoration protein (Dec) forms a capsid-stabilizing homotrimer that has an asymmetric tripod-like structure when bound to phage L capsids. The N-termini of the trimer subunits consist of spatially separated globular OB-fold domains that interact with the virions of phage L or the related phage P22. The C-termini of the trimer form a three-stranded intertwined spike structure that accounts for nearly all the interactions that stabilize the trimer. A Dec mutant with the spike residues 99-134 deleted (Dec 1-98 ) was used to demonstrate that the stable globular OB-fold domain folds independently of the C-terminal residues. However, Dec 1-98 was unable to bind phage P22 virions, indicating the C-terminal spike is essential for stable capsid interaction. The full-length Dec trimer is disassembled into monomers by acidification to pH <2. These monomers retain the folded globular OB-fold domain structure, but the spike is unfolded. Increasing the pH of the Dec monomer solution to pH 6 allowed for slow trimer formation in vitro over the course of days. The infectious cycle of phage L is only around an hour, however, implying Dec trimer assembly in vivo is templated by the phage capsid. The Thermodynamic Hypothesis holds that protein folding is determined by the amino acid sequence. Dec serves as an unusual example of an oligomeric folding step that is kinetically accelerated by a viral capsid template. The capsid templating mechanism could satisfy the flexibility needed for Dec to adapt to the unusual quasi-symmetric binding site on the mature phage L capsid.

Geometrically asymmetric pressure sensor with precious few electrodes achieves precise detection of both position and strength by machine learning

Wearable devices that can accurately detect the position and strength of applied pressure have been a focus of attention in the field of flexible tactile sensing. However, conventional sensor arrays with multiple units have many challenges in practical applications due to their complex circuit layouts and poor robustness, etc., which could be solved by minimizing the number of connecting electrodes among each unit. Herein, a pressure sensor with geometrically asymmetric structure has been proposed to remove all the connected inner electrodes. The sensor is constructed by three pieces of fabrics, with one insulating fabric sandwiched by two conductive fabrics coated with carbon nanotubes. Notably, the bottom fabric is specially designed in an optimized asymmetric shape with a rectangular hole located off-center of the square, assisting the sensor to generate different resistive and capacitive responses among pre-divided nine areas through only three external electrodes. Compared to conventional resistive sensor arrays, the number of electrodes is reduced by 83.3 %. With the assistance of machine learning, pressure-positioning with an accuracy of more than 98.1 % and precise strength measurements are achieved. Based on this sensor and the trained neural network, a two-level password system is constructed for more reliable role recognition. The asymmetric strategy proposed in this work provides a novel design for realizing precise detection of both position and strength using extremely simplified circuits, which has great potential in the field of wearable electronics.

Realizing enhanced photoresponse for self-powered broadband photodetector with asymmetric contacts based MoTe2/WSe2 van der Waals heterostructure

Self-powered photodetectors constructed by two-dimensional (2D) materials are significant in advanced science and technology with the growing focus on green and energy-efficient solutions. The built-in electric field in photodetector drives efficient separation and transfer of photogenerated electron–hole pairs. However, traditional metal-2D material-metal structures have more interface defects and larger Schottky barrier to hinder carrier transport. In this work, an asymmetric electrode material structure is constructed by adding graphene to provide a more efficient transport channel for photogenerated carriers. Graphene electrodes can play a variety of roles in efficient separation and collection of photogenerated carriers, reducing the barrier and improving the contact quality. Compared with the device using symmetric gold electrodes, the device prepared with asymmetric contacts show optimized performance. The MoTe2/WSe2 heterostructure exhibits brilliant self-powered photoresponse (a responsivity of 268 mA/W and a detectivity of 7.48*1011 Jones). And the rise/decay times are up to 145μs/125μs. Additionally, the detector has a broadband detection capability from visible light to near-infrared. These results provide a promising and simple way to fabricate a MoTe2/WSe2 self-powered broadband photodetector with high performance

Study on overburden rock structure characteristics and surrounding rock control technology of island working face

Based on the background of ZF2822 island working face in Xiagou Coal Mine, the characteristics of overburden fracture structure and surrounding rock control technology of island working face are explored by means of theoretical analysis, numerical simulation and field measurement. The results show that the ten morphological structures of the main roof fracture can be divided into four typical symmetrical structures and six asymmetric structures. According to the stress calculation formula of the key block B to the coal pillar, the stress characteristics of the 10 structures are classified and the risk classification is carried out. Four typical structures (a), (b), (c) and (d) were constructed to simulate the stress and strain characteristics of surrounding rock. The overlying strata subsidence displacement values of (a), (b), (c) and (d) at the roadway are 1.75 m, 1.5 m, 0.8 m and 0.6 m respectively, and the maximum stresses are 26.6 MPa, 19.5 MPa, 15.5 MPa and 10.0 MPa respectively, (a) > (b) > (c) > (d). Therefore, the surrounding rock control technology of gob-side roadway in island working face is put forward, the fracture structure of basic roof is determined and the reasonable applicable type of gob-side roadway is selected. The results after pressure relief show that the average daily microseismic energy of ZF2822 working face after pressure relief is 0.87 × 104 J, which is about 42.38% lower than that of April. The average stress is 1.13 MPa lower than that of the mining before the pressure relief area, and the decrease is 22.42%. It shows that the pressure relief has played a very good effect and provides a reference for the same type of mine.

Bio‐Inspired Design of 4D‐Printed Scaffolds Capable of Programmable Multi‐Step Transformations Toward Vascular Reconstruction

AbstractMany withered leaves or flowers spontaneously curl and transform from flattened structures into tubular constructs upon dehydration. Inspired by this phenomenon, an innovative strategy is developed to design stimuli‐responsive scaffolds that are capable of programmable transformation from flattened 2D constructs into various curled 3D tissue‐mimicking structures. Specifically, Janus‐structured scaffolds consisting of a passive layer of polycaprolactone and an active, triggered‐transforming layer of methacrylate gelatin/alginate hydrogel are 3D‐printed, which replicate the asymmetrical structure of leaves and enabled on‐demand transformation into desired curvatures and shapes through mimicking the regulation mechanism of leaves curling behavior by mesophyll matrix stiffness and vein patterns. Particularly for vascular reconstruction, multi‐step transformation scaffolds capable of primary 2D‐to‐3D transformation into tubular constructs induced by dehydration, and secondary transformation to adapt to the local intravascular geometry in vivo, are successfully developed. In addition, the oriented polycaprolactone layer of the printed scaffold can enable the adhesion, proliferation, and orientation of endothelial cells, thus demonstrating a novel strategy for the design of tissue‐engineered blood vessels. In general, this study provides an innovative design strategy for programmable biomaterials capable of shape morphing and adaptation in physiological conditions, thereby opening up a new avenue for the design of intelligent biomaterials in regenerative medicine.

Rotary pump using underwater electrical discharge

Powerful micropumps and water treatment are essential for biomedical applications using microfluidic circuits. Therefore, we propose a rotary pump using underwater electrical discharge for biomedical applications and elucidate its design concept. Specifically, we demonstrate that by applying high-voltage pulses repeatedly, the rotary device having an asymmetrical antenna structure can rotate with the maximum angular velocity of ∼25 rad s−1, and can produce a net flow with an average velocity of ∼3.2 mm s−1 along with an instantaneous maximum flow of ∼9 mm s−1. In addition, we explain our experimental results fairly well by proposing a simple model that considers the effects of asymmetricity and electric field strength with a steric effect. Our findings should contribute to the microfluidics for biomedical applications.

Circular Asymmetry of Ocean Mesoscale Eddy

Abstract In the global ocean, mesoscale eddies frequently deviate from a circular shape [circular asymmetry (CA)]. On average, anticyclonic eddies display slightly larger asymmetry than that of cyclonic eddies. Both types of eddies exhibit larger asymmetry during the periods of generation and extinction and smaller asymmetry during the intermediate periods. CA’s spatial distribution is dominantly controlled by the eddy’s rotational speed, radius, meridional displacement, and the background oceanic circulation. A decrease in rotational speed, an increase in radius, moving poleward (toward the equator), and/or moving into regions with smaller (larger) mean dynamic topography can enhance the asymmetry of anticyclonic (cyclonic) eddies. A weak advective nonlinearity of the eddy, represented by the ratio of the eddy’s rotational speed to its translation speed, also helps to enhance the asymmetry. Eddy with an asymmetric structure may be important for marine mixing processes. Significance Statement Mesoscale eddies are often assumed to be circular and symmetric. However, global observations reveal that eddies frequently diverge from a circular shape, displaying circular asymmetry (CA). The asymmetric structure of eddy is poorly investigated in the global ocean, especially in regions close to the strong oceanic currents. This study reported the evolution of eddy’s asymmetric structure and its spatial distribution. Statistical results show that CA is dominantly controlled by the eddy’s rotational speed, radius, meridional displacement, and the background oceanic circulation. This study provides a new perspective on the evolution of eddy’s structure and may be important in oceanic mixing.

Miniaturization of Power Dividers by Using Asymmetric CMRC Structures and QWLTs with Low-Cost Materials

Miniaturization of Power Dividers by Using Asymmetric CMRC Structures and QWLTs with Low-Cost Materials

Wake flow behind a transversely rotating sphere confined in a pipe at low Reynolds numbers

Direct numerical simulations are performed to investigate the flow past a transversely rotating sphere confined in a pipe. Three sphere diameters of d=0.2D, 0.5D, and 0.8D (D is the pipe diameter) and sphere Reynolds numbers (based on the sphere cross-sectional mean velocity and d) of Res=250–500 are considered. The simulation results highlight the combined effects of the blockage ratio (BR=d/D) and the nondimensionalized rotation rate Ω in the range of 0.2–1.2 on the wake flow. For each BR, with increasing Res and Ω, the wake flow undergoes a transition of the flow pattern from steady symmetry and unsteady symmetry to unsteady asymmetry. For BR=0.2, the variation in flow behaviors with Ω is similar to those reported in the previous studies on a transversely rotating sphere subjected to a uniform flow. With increasing BR from 0.2 to 0.5, the unsteady asymmetric wake flow is characterized by distorted near-wake vortex rings and a series of staggered inward rolled-up vortex arcs on the double streamwise vortex threads, which results in a low-frequency modulation of the unsteady wake flow. These rolled-up vortex arcs are connected with downstream quasi-streamwise vortices and form hairpin-like vortices. Dynamic Mode Decomposition (DMD) analysis shows that the distorted near-wake vortex rings are associated with the antisymmetric DMD mode at the adjacent frequency around the symmetric primary DMD mode. With BR=0.8, there is a drastic increase in the sphere's force coefficients. A more complex wake flow behavior is found due to a strong interaction between the wake flow and the near-pipe-wall flow. At Ω=1.2, dominant asymmetric flow structures are characterized by impingement of the vortex filaments. The low-frequency modulation of these wake structures is also analyzed using the DMD analysis.

Asymmetric development of overburden fracture and gas migration law for a goaf of entry formed by roof cutting

AbstractIn the study, a combined numerical simulation and on‐site monitoring method was used to analyze the asymmetric development characteristics of overlying rock fractures in a goaf under the condition of a goaf side entry formed by roof cutting and to explore the gas accumulation area in the goaf, achieving precise gas extraction from the goaf. The results demonstrates that a double‐balanced arch structure is formed under the condition of a goaf side entry formed by roof cutting, achieving safe retention of the roadway and showing the significance of the pressure relief effect of roof cutting. The collapse movement of the overlying rock on the roof‐cutting side is relatively advanced. The heights of the collapse zone on the roof‐cutting side and the uncut roof side are 28 and 24 m, respectively, and the development heights of the fracture zone are 37 and 42 m, respectively. The fault line on the roof‐cutting side gradually shifts toward the direction of the goaf, and the surface settlement and fracture development are relatively small. There is a clear asymmetric structure in terms of time effect, fault line, fracture zone height, and surface settlement compared to the uncut roof side. The gas is distributed throughout the entire goaf in the roof‐cutting and tunneling mode, and a high‐concentration gas accumulation area is formed near the open–off cut and working face on the high side of the fracture zone. Based on an actual situation, a method of drilling high and low positions in a fracture zone is proposed for extraction. Combined with on‐site monitoring, the goaf was no longer filled with gas during extraction, and the proportion of low‐concentration gas space considerably increased.

Tropical Cyclone Boundary Layer Asymmetries in a Tilt-Following Perspective

Abstract Previous observations and modeling studies showed that tropical cyclones (TCs) in a sheared environment develop an asymmetric boundary layer (BL). While the relationship between the BL asymmetries and environmental shear has been demonstrated, the exact cause of these BL asymmetries and the phase relationship between them are less well understood. In this study, we examine the dynamical processes leading to the asymmetric structure of the TC BL in a sheared environment using idealized, convection-permitting model simulations. Our results show that the emergence of the BL asymmetries is closely linked to the TC vortex tilt and rainband processes. Specifically, stratiform diabatic processes in the downtilt-left region result in midlevel descending inflow, which brings midtropospheric, low-θE air toward the BL and forms a surface cold pool in the downtilt-left quadrant. This descending inflow also advects high absolute angular momentum inward, redistributing the vertical vorticity and causing a storm-scale tangential wind acceleration within the downtilt-left quadrant. As the BL low-θE air advances inward, it becomes supergradient and decelerates radially, forming BL outflow in the uptilt-left quadrant. The outflow advects positive relative vorticity uptilt, forming an elliptic BL vorticity and circulation structure. As the tilted TC vortex and the accompanying rainband precess cyclonically over time, the above sequence of events and the resultant BL asymmetries also precess cyclonically, maintaining a quasi-stationary configuration relative to the vortex tilt. These results suggest that the primary organizing factor of the boundary layer asymmetries is the tilted vortex structure and not strictly the environmental shear direction.

Bayesian modeling of the Gompertz curve for meat quails growth data considering different error distributions

This study applied the Gompertz model to quail growth data, assuming symmetric and asymmetric homoscedastic and heteroscedastic error distributions (Normal, t-Student, Skew normal, and Skew t), undera Bayesian framework. Model selection criteria included the Bayesian Deviance Information Criterion (DIC) and the analysis of residual standard deviation (σ), as well as graphical assessment of the fit. For both homoscedastic error structures (males: DIC=7.186; σ=10.73) and (females: DIC=5.572; σ=11.88) as well as heteroscedastic structures (males: DIC=6.493; σ=0.795) and (females: DIC=4.405; σ=0.824), the best fits were obtained by considering the Skew t distribution for errors. In homoscedastic fits, significant residual asymmetry (λ) was observed only for female quails (CI(λ)=[-8.039;-0.340]), whereas in heteroscedastic fits, the parameter was not significant for both sexes. Additionally, heteroscedasticity (δ) captured in the fits was significant for both sexes (males: CI(δ)=[1.66;2.13] and females: CI(δ)=[1.80;2.26]). Understanding animal growth is crucial for optimizing management and feeding practices, reducing time and costs in production. In this case, the use of nonlinear models considering heteroscedastic and/or asymmetric residual structures contributes to greater accuracy in decision-making.

A novel mixture rule for multi-layer plates with thermal bending and twisting effects applied to FOCoS devices

A novel approach for reducing the simulation time of highly complex fan-out chip on substrate (FOCoS) structures was developed. Based on the Uflyand-Mindlin plate theory and using the multi-layer plate technique, we have successfully obtained the effective elastic and thermal expansion constants incorporating with the effects of bending and twisting for a two-layer plate structure. The newly developed approach was then applied to simulate, by the finite element method, a variety of cases commonly encountered in practice, such as a slightly asymmetric substrate, an extremely asymmetric rigid structure, and an extremely asymmetric soft structure, with a maximum error of only 3 μm. Remarkably, applying the present mixture rule to simulate the complete FOCoS structure has resulted in an error of only 10 μm with up to 75 % of reduction in the number of mesh elements. The present approach, hence, provides an effective and reliable means for simulating the ever increasingly important FOCoS structures in the rapidly evolving information technology applications.