Efficient Analytical Solution Research Articles

A modified solution for evaluating the sliding direction and stability of 3D slopes

In this paper, we present an improved and efficient analytical solution for evaluating the stability of 3D slopes based on the minimum potential energy method. The equilibrium equation of the landslide is employed to determine the shear stress on the failure surface, and a new formulation for shear potential energy is introduced. The vector-combined direction of the force acting on the landslide is considered equivalent to the sliding direction (SD). Two classic examples are reanalyzed to validate the accuracy and performance of the proposed method. The analysis demonstrates that the obtained results align well with reference solutions and tend to be conservative in engineering applications. Parameter investigations are conducted to explore the influences of SD, shear strength parameters, slope angle, and soil unit weight on the safety factor (SF) of 3D slopes. The findings indicate that a larger SD corresponds to a less stable slope. Finally, the effects of optimization parameters, landslide volume, and surcharge on the location of the failure surface, SF, and SD are also examined. The developed analytical approach offers a novel and straightforward solution with sufficient accuracy and eliminates the need for iteration in determining the SF, making it a valuable complementary method for assessing the stability of 3D slopes.

Fractional view analysis of the diffusion equations via a natural Atangana-Baleanu operator

In this article, the natural transform decomposition method (NTDM) is applied to evaluate the fractional-order diffusion equations utilizing the Atangana-Baleanu operator. The method offers an efficient analytical solution by combining the natural transform methodology with a decomposition strategy. The suggested method incorporates non-local effects and memory qualities introduced by the Atangana-Baleanu operator while demonstrating adaptability and applicability to various fractional-order diffusion equations. In comparison to previous analytical techniques, the NTDM has demonstrated the lowest calculation volume and the highest rate of convergence. The obtained results are presented illustratively by using graphical simulations. The suggested method is therefore easily adaptable to other non-linear issues. The NTDM includes comparative analysis capabilities, practicality, and accurate results for researchers and practitioners. In order to solve fractional-order phenomena, particularly diffusion equations, NTDM is regarded as the best analytical technique.

An efficient vectorization solution to cutting dynamics modeling for face-hobbing of hypoid gears

The cutting vibration generated by cutting forces is critical to the accuracy and efficiency of face-hobbed hypoid gears. However, few relevant works are currently on cutting dynamics modeling for the face-hobbing process due to its complex spatial relationships. This study developed an efficient cutting dynamics model considering comprehensive characteristics of the non-generated face-hobbed hypoid gear machining process. A concise and efficient analytical solution to cutting dynamics modeling for face-hobbing of hypoid gears is obtained by using a vectorization method to characterize chip regenerating mechanisms. The machine tool vibration caused by cutting forces is predicted and validated according to an industrial case study. The proposed cutting dynamic model is 36.11% more efficient than the semi-analytical dynamic model derived from other cutting force prediction methods, and its accuracy is also improved with an error of less than 15%. Furthermore, the proposed vectorization solution to face-hobbing cutting dynamics modeling will be a practicable and effective method for analyzing the chatter in the machining of complex surfaces.

Mapping the Unresolved Tidal Resource in Estuaries

Estuaries are ideal locations for extracting tidal energy and yet the global resource appears poorly mapped. Estuaries typically have high tidal ranges and strong tidal currents, due to amplification processes; and this resource is juxtaposed to industrial/residential areas with electricity demand [1]. For example, 22 of the 32 largest cities in the world are adjacent to estuaries [2], and UK estuaries collectively worth over £5.5b to UK economy with >1/6th of the population and ~1b tonnes of cargo traded at their ports [3]. 
 Mapping the tidal energy resource is challenging due to the sensitivity of ocean-model simulated currents to the model resolution [4]. Both tidal amplitude and currents are heavily modified in estuaries [5]. Whilst many estuary-specific resource modelling studies have been achieved (e.g. [6]), no large-scale future tidal energy resource mapping project has been undertaken – likely due to computational cost (e.g. [4]). It is therefore unrealistic to hydrodynamically model all global estuarine systems, to resolve current and future changes to the tidal energy resource; instead we aim to apply a simplified analytical technique that could be calibrated and validated by the new NASA SWOT satellite mission (https://swot.jpl.nasa.gov/), as well as citizen science. Estuarine tidal dynamics have been observed to change rapidly due to changes in river-flow climatology and bathymetry (e,g. dredging); see [7]. Indeed, climate change driven impacts to tidal dynamics (sea-level rise and altered riverine climatology), alongside anthropogenic driven morphodynamical changes and interactions with future tidal dynamics, could increase the future estuary tidal resource [1].
 Three physical processes drive mean tidal amplification excluding over-tides): (1) Funnelling - concentration of the tidal energy flux with reducing width; (2) Resonance - when estuary length-scales are close to the natural period of the basin; and (3) Shoaling (reduction in the propagation speed of the tidal wave resulting in an increase in amplitude). Each of these three processes are analytical solved, and a 1D analytical model applied to demonstrate the resource. Sea-Level Rise (SLR) scenarios are included to show SLR can modify estuarine tidal dynamics through both estuarine geometry (e.g. increasing resonance) and boundary conditions (global mean sea-level rise increasing boundary conditions). A normalised result of the analytical approach is shown in Figure 1, which demonstrates a computationally efficient analytical solution that includes future changes to tidal dynamics. 
 Our research will apply this analytical solution to the Bristol Channel, a region soon-to-be heavily instrumented as part of the “Cal-Val phase of the NASA SWOT project (https://projects.noc.ac.uk/swot-uk/swot-uk-bristol-channel). Finally, we will develop a technique to bring together digital observations from citizens and sensor networks (see https://digitalenvironment.org/), to validate our simplified approach and resolve the tidal energy resource.

Rate-dependent JKR-type decohesion of a cylindrical punch from an elastic substrate

Recently published experimental data on non-quasistatic detachment of a flat-ended cylindrical punch from an adhesive rubber layer are analyzed in the framework of axisymmetric rate-dependent JKR-type model. The functional dependence of the work of adhesion on the velocity of the contour of contact area is assumed according to the known Gent–Schultz model. The evolution of the variable contact radius as a function of the punch displacement is described by a first-order ordinary differential equation, which possesses the localization property for its solutions, meaning that the detachment occurs at some nonzero contact radius. To facilitate the model fit to experimental force-displacement curve, a computationally efficient analytical approximate solution is suggested. A parametric analysis of the basic case (when the rubber layer is approximated by an elastic half-space) is presented.

Efficient Analytical Model for Time-Dependent Behavior of Layered Functionally Graded Plates with Viscoelastic Interlayers

Layered functionally graded (FG) plate plays an important role in engineering constructions. In this work, an efficient analytical solution is proposed to investigate the time-dependent characters of layered FG plate with viscoelastic interlayers, in which each FG layer is described by the elasticity theory, and the interlayer viscoelasticity is simulated by the generalized Maxwell model. The constitutive equations in the interlayer are simplified, and then, the analytical solutions of stresses and displacements for the layered FG plate are solved by virtue of the efficient recursive matrix method. Some examples are analyzed to investigate the influences of geometric and material characteristics on the long-term behaviors for the FG plate.

A Method to Improve Computational Efficiency of Productivity Evaluation with Rectangular Coalbed Methane Reservoir

Computational efficiency is the key factor to be considered in the productivity evaluation of rectangular coalbed methane reservoir. There are three main factors affecting the calculation speed: the nonlinearity of the material balance equation of coalbed methane reservoir, the poor conductivity of fractures cannot be considered as infinite conductivity fractures, and the Duhamel convolution is needed in history fitting and boundary image inversion. At present, there is no method to quickly evaluate the productivity of finite conductivity fracture model in rectangular coalbed methane reservoir. Diffusion equation of matrix is generated by the Fick diffusion law. The Darcy seepage law is used to build the seepage equation of fractured system in coalbed methane reservoir. In order to transform the calculation result of infinite conductivity fracture into finite conductivity fracture, fracture conductivity factor is employed in this paper. The applicability of fracture conductivity factor in the whole production process is clarified. It is clear that the factor is prone to calculation errors when the time is small, and the calculation fluctuates greatly. According to the characteristics of the Riley method and discrete method, an accurate and efficient analytical solution calculation process is designed. This will make the calculation results accurate. A production evaluation method of rectangular coalbed methane reservoirs with fractured vertical well and finite conductivity fracture is proposed. The purpose of quickly and accurately predict well production capacity is reached. The geological parameters are recombined, and new coalbed methane reservoir flow parameters are defined. Through parameter sensitivity analysis, the influence of different flow characteristic parameters on gas production is clarified. The dimensionless transfer constant and dimensionless storage capacity affect the appearance time of desorption and diffusion and the storage capacity of the fracture system, respectively. The dimensionless desorption constant describes the strength of desorption and diffusion. The influence of fracture conductivity factor on production is studied. It is clarified that its impacts are different in the early stage and the later stage of production. There is a limit to the fracture conductivity factor. When the limit is exceeded, the fracture conductivity factors no longer affect the production of a single well. The findings of this study can understand the percolation stage of finite conductivity fractured wells with rectangular coalbed methane reservoir and can also guide fracturing design and writing in the field. The research results enrich the productivity evaluation model of coalbed methane reservoir. In the end, a set of production evaluation method is put forward suitable for the well in rectangular coalbed methane reservoirs with fractured vertical well and finite conductivity fracture. In this paper, the influence of fracture conductivity on single well productivity in rectangular coalbed methane reservoir is quantitatively evaluated for the first time. By improving the calculation method and optimizing the calculation path, the productivity evaluation calculation speed of finite conductivity fractured wells in rectangular coalbed methane reservoir is optimized without affecting the calculation accuracy. The new method can be applied directly to productivity evaluation software, which has the significance of popularization.

Coupled whole-tree optimality and xylem hydraulics explain dynamic biomass partitioning.

Trees partition biomass in response to resource limitation and physiological activity. It is presumed that these strategies evolved to optimize some measure of fitness. If the optimization criterion can be specified, then allometry can be modeled from first principles without prescribed parameterization. We present the Tree Hydraulics and Optimal Resource Partitioning (THORP) model, which optimizes allometry by estimating allocation fractions to organs as proportional to their ratio of marginal gain to marginal cost, where gain is net canopy photosynthesis rate, and costs are senescence rates. Root total biomass and profile shape are predicted simultaneously by a unified optimization. Optimal partitioning is solved by a numerically efficient analytical solution. THORP's predictions agree with reported tree biomass partitioning in response to size, water limitations, elevated CO2 and pruning. Roots were sensitive to soil moisture profiles and grew down to the groundwater table when present. Groundwater buffered against water stress regardless of meteorology, stabilizing allometry and root profiles as deep asc.30m. Much of plant allometry can be explained by hydraulic considerations. However, nutrient limitations cannot be fully ignored. Rooting mass and profiles were synchronized with hydrological conditions and groundwater even at considerable depths, illustrating that the below ground shapes whole-tree allometry.

Modeling colonization rates over time: Generating null models and testing model adequacy in phylogenetic analyses of species assemblages.

A central theme connecting macroevolutionary processes to macroecological patterns is the shaping of regional biodiversity over time through speciation, extinction, migration, and range shifts. The use of phylogenies to explore the dynamics of diversification due to variation in speciation and extinction rates has been well-developed and there are established methods for inferring speciation times from phylogenies and generating its null distributions (as represented by node heights on molecular phylogenies). But inferring colonization events from phylogenies is more challenging. Unlike speciation events, represented by nodes, colonization events could occur at any point along a branch connecting species in the assemblage to the regional pool. We account for uncertainty in identification of colonization lineages and timing of colonization events by using an efficient analytical solution to inferring the distribution of colonization times from an assemblage phylogeny. Using the same solution, we efficiently derive the null distribution of colonization times, which provides us with a general approach to testing the adequacy of a model to describe colonization events into the assemblage. We illustrate this approach by demonstrating how the movement of squamate lineages into Madagascar has been uneven over time, peaking in the early Cenozoic when ocean conditions favored colonization.

Experimental and numerical analysis on vibration of plate with multiple cutouts based on primitive cell plate with double cutouts

An efficient analytical solution is presented in this paper to model the bending vibration of plates with any numbers of cutouts and under arbitrary boundary conditions. Especially we study the efficiency of the model while explore the influence of increasing number of cutouts and increasing area of cutouts on the eigenpairs of the plate. In the first step, the model of the plate with cutouts is decomposed into certain amount of primitive cells with double cutouts and solved based on the Chebyshev-Lagrangian method. The two advantages of this modeling method lie in offering a flexible assemble strategy of primitive cell plates to form plate in various shapes and with arbitrary numbers of cutouts while bringing substantial convenience in calculation owing to the uniformly expanded double Chebyshev series terms for the displacement of each primitive cell plate. Secondly, the experimental and numerical validations are performed and consequent efficiency and convergence analysis is discussed. The results show that the current model presents potential robust in computation efficiency. At last, the influence of several key parameters on eigenpairs of plate with cutouts is investigated, including the boundary restraint of the plate, the number of cutouts and the size of cutouts.

Comparison research of ship-to-ship hydrodynamic interaction in restricted water between TEBEM and other computational method

In this study, we systematically analyze the hydrodynamic interaction of two ships navigated parallel to each other in shallow water using the Taylor Expansion Boundary Element Method (TEBEM). As a low ship speed is assumed, the influence of the free-surface condition is neglected in the initial simulation. The influence of various parameters, including ship speed, water depth, and lateral distance between ships, on interaction force and moment are analyzed. Validation of the results using experimental data, boundary element method results, and numerical simulation results obtained using the Star CCM + commercial computational fluid dynamics software confirms that the TEBEM results are similar to those obtained using the high-order boundary element method and better than those obtained using the constant panel method. Further simulations assuming a free-surface effect are carried out using computational fluid dynamics calculations. Our results provide a high-precision and efficient analytical solution to the effects of viscosity, including the lifting force caused by asymmetric flow and free surfaces, in ship-to-ship interaction in restricted water and deeper insight into the influence on the results of the factors neglected by the potential flow simulation.

Enhancing supply chain production-marketing planning with geometric multivariate demand function (a case study of textile industry)

In this paper, a multi-period, multi-product, multi-site, multi-sales channel aggregate production planning problem including ordering preferences is presented in an integrated two-echelon supply chain to avoid the sub-optimality caused by separate, sequential decisions of production and the marketing/retailing chain. Each customer demand class is affected by price, marketing expenditures and product quality involving customer willingness-to-pay. In addition, the immigration of customers between submarkets (i.e. cannibalization) is considered in the market-segmented environment due to imperfect segmentation. This research develops a geometric programming model to formulate the issue of joint price differentiation and multi-site aggregate production planning decisions by maximizing the total profit of the supply chain. To tackle the model and obtain solutions, we tailor an efficient analytical solution procedure to convert the original highly non-linear programming model into a convex programming equivalent. Finally, a numerical study of garment supply chain is presented to demonstrate the performance of the model and solution approaches. The research findings indicate a positive relationship between the scaling constant of price-dependent demand and the total profit rate. Moreover, as price gaps grow, the utility of price differentiation is decreased.

Hybrid metal-composite conical tubes for energy absorption; theoretical development and numerical simulation

Abstract In this paper, the crushing behavior of hybrid metal-composite conical tube under dynamic loading is studied. An efficient analytical solution for FML conical tubes consist of any number of metal and laminated composite layers is developed. In the analytical analysis, the mean collapse load of structures subjected to axial loading was predicted while its accuracy is validated via experimental tests and numerical simulation. Numerical simulation of the structure is also done using explicit dynamic finite element software in order to investigate the effects of different parameters on crushing characteristics of various structures. On the other hand, new values for failure energies of fiber reinforced composites are proposed in the failure evolution criteria in the finite element model. It leads to good agreement between FE simulations and experimental test other. Moreover, a comprehensive parametric study has been done in order to investigate the effect of various parameters including semi-apical angle, ply pattern of laminated composite, diameter of tube, wall thickness and material properties of tubes on energy absorption capacity and crashworthiness characteristics of various considered specimens. Finally, the crashworthiness capacity (SEA and CLE) with respect to semi-apical angle and ply pattern of laminated composite optimized in order to obtain a desired collapse mode. Based on the obtained results the optimized structure is determined. Also, crushing behaviors of various specimens predicted beyond the tested configurations using response surface methodology (RSM).

Analytical and experimental study of the ultimate strength of delaminated composite laminates under compressive loading

An improved and efficient analytical solution is proposed to predict the behaviour and the ultimate strength of composite laminates containing delamination. For this purpose, a Layerwise Higher order Shear Deformation Theory based on polynomial shape functions using Rayleigh–Ritz approximation technique are utilized. The ultimate strength and initial stages of propagation of the delaminated zone are determined using fracture analysis. The results demonstrate continuous functions to calculate in-plane and out-of-plane displacements, rotations, strains, stresses, energy and load carrying capacity under in-plane compressive loading. Moreover, a significant experimental investigation has been carried out for specimens with cross-ply stacking sequence and various material properties, geometries, delamination type and boundary conditions. The experimental study includes fabrication process, tensile tests to determine material properties and buckling test of the defected specimens.

Efficient crushable corrugated conical tubes for energy absorption considering axial and oblique loading

Thin-walled structures are of much interest as energy absorption devices for their great crashworthiness and also low weight. Conical tubes are favorable structures because unlike most other geometries, they are also useful in oblique impacts. This paper investigated the effect of corrugations on energy absorption characteristics of conical tubes under quasi-static axial and oblique loadings. To do so, conical tubes with different corrugation geometries were analyzed using the finite element explicit code and the effects of corrugations on initial peak crushing force and specific energy absorption were studied. The finite element model was validated by experimental quasi-static compression tests on simple and corrugated aluminum cylinders. An efficient analytical solution for EA during axial loading was also derived and compared with the FEM solution. The crushing stableness was analyzed using the undulation of the load-carrying capacity parameter and it was shown that corrugations made collapsing mode, more predictable and controllable. The findings have shown that corrugated conical tubes have much better energy absorption characteristics compared with their non-corrugated counterparts. It was also discovered that during oblique loadings, introducing corrugations can significantly increase the specific energy absorption compared with simple cones.

Vibration Behavior of Nanocomposite Plate Reinforced by Pristine and Defective Graphene Sheets; an Analytical Approach

Free vibration characteristics of polymer composite plates reinforced by graphene nanosheets employing the Eringen nonlocal elasticity theory were investigated. Theoretical formulations are derived based on Hamilton’s principle implementing linear orthotropic constitutive equations of lamina while the behavior of nanostructure points affected by all other nonlocal points is also taken into account. For obtaining the mechanical properties, a new modified Halpin–Tsai model is employed. Governing equations are solved by developing an efficient analytical solution. The accuracy of the presented method is examined, by comparing the results with literature in which a good agreement is observed. Effects of different boundary conditions, volume fraction, graphene sheets orientation angle and Eringen nonlocal parameter on frequency of nanocomposite are analyzed. Effects of the presence of vacancy defects in the nanosheet on the behavior of reinforced composites were also studied. The results illustrate that by increasing the nonlocal parameter the natural frequency showed a decreasing trend while by increasing the graphene sheet’s volume fraction, natural frequencies significantly increased. It could be concluded that the orientation angle variations in graphene sheets, did not play an important role on the natural frequency of nanocomposite as well as degradation of properties resulted in from vacancy defects.

Energy absorption analysis of a novel foam-filled corrugated composite tube under axial and oblique loadings

Composite thin-walled structures are of much interest in differnet applications as well as energy absorption devices for their great crashworthiness and light weight. In this paper, a new corrugated composite cylindrical tube has been introduced in order to improve crashworthiness along with a stable crushing. In cylindrical composite tubes, the effects of corrugations regarding charactristics of energy absorption have underwent quasi-static axial and oblique loading investigations. For this reason, composite cylindrical tubes with different corrugation geometries were analyzed using finite element explicit code and the effects of corrugations on crush force effiency and specific energy absorption were comperhensively studied. The finite element model has been validated by experimental quasi-static compression tests. An efficient analytical solution for SEA during axial loading has been also derived and compared with FEM solution. Furthermore, a comparison of empty and foam-filled corrugated composite tubes has been done. Based on the obtained results, generating corrugated surfaces on tubes improved the crush force efficiency significantly in both axial and oblique crushings. Performing a parametric study on geometrical corrugation parameters of tubes has been indicated that the energy absorption of these structures depends strongly on the corrugation parameters. Furthermore the absorbed energy has been increased by using foams in both axial and oblique crushing. SEA increases by increasing the foam density while the CFE decreases.

Solution of fuzzy fractional differential equations using homotopy analysis method

This paper presents an efficient analytical solution for fuzzy fractional differential equations (FDEs) using homotopy analysis method. HAM is the generalization of various methods like homotopy perturbation method and Adomian decomposition method. The main advantage is its rapid convergence to the solution without using perturbation or restrictive assumptions. Illustrative numerical example are presented for the proposed method.

Time Optimal Contouring Control of Industrial Biaxial Gantry: A Highly Efficient Analytical Solution of Trajectory Planning

Contouring control is an important issue in industrial applications. For better product quality and higher productivity, achieving higher contour tracking performance with feed rate as large as possible is essential. However, most existing research has focused on improving contour tracking accuracy only. The potential instability issue due to control input saturation under high feed-rate operations has been largely ignored. In this paper, the minimum time trajectory planning problem is formulated as a step toward solving this practically important problem. A novel back and forward check algorithm is developed to take into account physical constraints of the system when generating the desired trajectory to be followed. Compared with traditional numerical searching methods, the proposed algorithm is very computationally efficient because the optimal solution at each node can be obtained by several analytical equations directly. Moreover, the algorithm can incorporate some velocity-dependent constraints easily. Planning a Lissajous curve is used as the case study to verify the optimality and computational efficiency of the proposed algorithm. Experiments are conducted on an industrial biaxial gantry. Several implementation issues on tracking complicated contours are discussed. Experimental results indicate that the proposed trajectory planning approach enables simultaneous achievement of high contouring tracking accuracy and higher feed rate or shorter operation time.

The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation

In this paper, an efficient analytical solution method, namely, multi-level residue harmonic balance, is introduced and developed for the nonlinear vibrations of multi-mode flexible beams on an elastic foundation subject to external harmonic excitation. The main advantage of this solution method is that only one set of nonlinear algebraic equations is generated in the zero level solution procedure while the higher level solutions for any desired accuracy can be obtained by solving a set of linear algebraic equations. In other words, the computation effort to find more accurate nonlinear solutions is much less. In this paper, a multi-mode formulation, which represents the nonlinear beam vibration, is derived and set up. Then, the solution procedures are developed for obtaining the nonlinear multi-mode solution. The results from the multi-level residue harmonic balance method agree well with those from a numerical integration method. The effects of various parameters such as vibration amplitude, foundation modulus coefficient, damping factor and excitation level etc., on the nonlinear behaviors are examined. A convergence study is also performed to verify the solutions. The stability analysis is conducted using the virtue of Floquet theory and steady-state solutions are investigated.