Residue Regulating Homotopy Method for Strongly Nonlinear Oscillator

It is highly desired yet challenging to obtain analytical approximate solutions to strongly nonlinear oscillators accurately and efficiently. Here we propose a new approach, which combines the homtopy concept with a “residue-regulating” technique to construct a continuous homotopy from an initial guess solution to a high-accuracy analytical approximation of the nonlinear problems, namely the residue regulating homotopy method (RRHM). In our method, the analytical expression of each order homotopy-series solution is associated with a set of base functions which are pre-selected or generated during the previous order of approximations, while the corresponding coefficients are solved from deformation equations specified by the nonlinear equation itself and auxiliary residue functions. The convergence region, rate and final accuracy of the homotopy are controlled by a residue-regulating vector and an expansion threshold. General procedures of implementing RRHM are demonstrated using the Duffing and Van der Pol-Duffing oscillators, where approximate solutions containing abundant frequency components are successfully obtained, yielding significantly better convergence rate and performance stability compared to the other conventional homotopy-based methods.

Acoustic and Dynamic Response of Unbaffled Plates of Arbitrary Shape

In this study, a method for determining the effects of fluids on the dynamic characteristics of an aerospace structure and the response of the structure when it is excited by the acoustical loads produced during a rocket launch, has been developed. Elevated acoustical loads are critical in the design of large lightweight structures, such as solar arrays and communication reflectors, because of the high acceleration levels. The acoustic field generated during rocket launch can be considered as a diffuse field composed of many uncorrelated incident plane waves traveling in different directions, which impinge on the structure. A boundary element method was used to calculate the pressure jump produced by an incoming plane wave on an unbaffled plate and the fluid–structure coupled loads generated through plate vibration. This method is based on Kirchhoff’s integral formulation of the Helmholtz equation for pressure fields. The generalized force matrix attributed to the fluid loads was then formulated, taking the modes of the plate in vacuum as base functions of the structural displacement. These modes are obtained using a finite-element model. An iteration procedure was developed to calculate the natural frequencies of the fully coupled fluid–plate system. Comparison of the results obtained using the proposed method with those of other theories and experimental data demonstrated its efficiency and accuracy. The proposed method is suitable for analyzing plates of arbitrary shape subjected to any boundary conditions in a diffuse field for low to medium values of the frequency excitation range.

Tristable Properties in Electrostatically Actuated Initially Curved Coupled Micro Beams

A limit point behaviour analysis of a metastructure, composed of two double clamped, initially curved beams, coupled via a rigid truss at their respective centres, is carried out when subjected to a distributed electrostatic load. The analysis is based on a reduced order (RO) model resulting from Galerkin’s decomposition, with symmetric buckling modes taken as the base functions, for either beam. All solutions employed the implicit arc-length “Riks” method to accommodate for winding equilibrium paths, while validation of the said results were carried out against finite differences (FD) direct solutions. In addition, local stability analysis via the energy method, conducted on the primary beam was instrumental in clarifying the role of the various extremum points by characterising which branches are stable, and which are not. The combined analysis has shown that the driving beam, which directly encounters the load, is able to possess bistable as well as tristable properties, provided that the metastructure meets certain geometrical parameters. Several variations of tristability are disclosed in the study. The analysis indicates that a model with at least three degrees of freedom (DOF) is needed to predict such configurations, as well as the various critical thresholds, with reasonable errors of around one percent when compared against FD. In so doing, the model can be used to provide static characterisation of the structure.

On applicability of MQ-RPIM and MLPG meshless methods with 3D extended-enriched base functions for estimation of mode I stress intensity factor and fatigue crack growth in cyclic tensile and bending load of an un-notched and notched shaft

In this study, a numerical meshless method is used to solve the weak form of the linear elastic equations in solid mechanics. Evaluation and comparison of the numerical meshless methods have been carried out via the radial point interpolation meshless method with multi-quadrics base functions (MQ-RPIM) and meshless local Petrov-Galerkin method (MLPG). Using these two methods, stress intensity factors in an elastic medium containing geometric discontinuities and cracks are estimated based on tensile and bending cyclic loading. The analysis domain has been identified via three-dimensional modeling of the notched and un-notched shafts with an initial surface semi-elliptical crack subjected to tensile or bending cyclic loadings. To enhance the accuracy of calculations, the RPIM meshless method is applied using polynomial and extended-enriched 3D base functions. Shape functions have been developed using standard and optimal parameters and values with Mono-Objective Function in PSO algorithm. In the MLPG meshless method with the extended-enriched functions, discretization is performed via direct and penalty factor methods, to reach more efficient results and meet the boundary conditions. Efficiency comparison of the selected numerical methods with the experimental findings and the numerical analysis of finite elements method indicates that in comparison with the MLPG method, MQ-RPIM enriched meshless method can be utilized with fewer nodes in the analysis domain while reaching the accuracy and convergence with lower stress intensity factors and gentler slope. However, the processing time of the MLPG meshless method is lower than that of the other methods.

RESEARCH OF DATA TYPE CLASSIFICATION METHODS WHEN DEVELOPING COMPUTER ENGINEERING SOFTWARE

The paper deals with the problems of increasing the efficiency of software development, in particular, the issue of reducing the time for developing programs and using automated synthesis of programs, which will avoid the revision of the original product. The software should be tested along with other system components in all combinations that may occur. Testing is time-consuming because hidden bugs are revealed through unexpected interactions between software components. With structural analysis, data flow diagrams are not the end result, they are a developer tool. First, diagrams are built, and then mechanisms are developed to ensure the required system behavior. A graphical approach to solving the problem of automation of software development is being developed, based on the involvement of visual forms of program presentation. For any program object, you can select a finite number of states in which it is at each moment of time. The program progress is associated with the transition of an object from one state to another. The graph replaces the textual form of the description of the program algorithm, while the visual representation of the algorithm is realized. The specification of data structures, as well as the setting of intermodular interfaces according to data, is separated from the description of the structure of the algorithm and controls. Basic modules and data types are used. Basic modules are local calculable functions, on the basis of which all other technology objects are generated. Data types describe the syntactic and semantic aspects of constructing data used in base functions. Algorithms for finding routes on directed graphs are considered. When defining routes from the root vertex to the final ones, the properties of the algebra of three-valued logic were used. Based on the considered approach, as well as taking into account its shortcomings, a method for classifying data types was proposed, based on the implementation of a partial enumeration of the routes of the graph of program links and a method for designing software based on it, taking into account minimizing the time and cost of the project. Keywords: software, computer engineering, information systems, components, partial enumeration of graph routes, development costs.

Thermo – mechanical model of concrete pavement in hardening phasis

This paper is focused on the analysis of concrete pavements using finite element method (FEM). Specifically, it deals with the analysis of temperatures in the initial phasis of hardening and their influence on mechanical behavior of concrete pavement. High temperatures from hydration and climatic conditions in the early phase of concrete hardening co-operate and may initiate the formation of a network of micro-cracks on the surface of the concrete slab. The resulting temperatures (from hydration and climate) can theoretically be positively influenced by determining the start of concreting, so that the maximum temperatures do not meet at the same time. However, from a practical point of view the use of retarders is more realistic. Another possibility is to reduce the hydration heat by changing the composition of the concrete mixture (amount of cement, type of cement, use of alternative binders). Based on the knowledge of the material composition of the concrete and the specific temperature behavior during the concrete laying, it will be possible to predict the durability of concrete pavement in the future. Using weak formulation FEM model with quadratic base functions, the 2D heat transfer model was created. Boundary conditions were determined from experimental measurement on highway D1 in the Czech Republic. When this model was fitted to experimental data, the 3D coupled thermo - mechanical model was created. Soil and concrete elastic material characteristics had been taken over from Czech technical norms. Soil was modelled as Winkler-Pasternak 2D plate. Parameters c1 a c2 were assessed from comparison with 3D model with soil modelled as multiple layer system.

A hybrid $$ H ^1\times H (\mathrm {curl})$$ finite element formulation for a relaxed micromorphic continuum model of antiplane shear

One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to $$ H ^1$$ H 1 , such that standard nodal $$ H ^1$$ H 1 -finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces $$ H ^1$$ H 1 and $$ H (\mathrm {curl})$$ H ( curl ) , demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates.

Approximate Solution of Bratu Differential Equations Using Trigonometric Basic Functions

In this paper, I have proposed a method for finding an approximate function for Bratu differential equations (BDEs), in which trigonometric basic functions are used. First, by defining trigonometric basic functions, I define the values of the transformation function in relation to trigonometric basis functions (TBFs). Following that, the approximate function is defined as a linear combination of trigonometric base functions and values of transform function which is named trigonometric transform method (TTM), and the convergence of the method is also presented. To get an approximate solution function with discrete derivatives of the solution function, we have determined the approximate solution function which satisfies in the Bratu differential equations (BDEs). In the end, the algorithm of the method is elaborated with several examples. In one example, I have presented an absolute error comparison of some approximate methods.

Summation-Integral Type Operators Based on Lupas-Jain Functions

We introduce a genuine summation-integral type operators based on Lupaş-Jain type base functions related to the unbounded sequences. We investigated their degree of approximation in terms of modulus of continuity and ????-functional for the functions from bounded and continuous functions space. Furthermore, we give some theorems for the local approximation properties of functions belonging to Lipschitz class. Also, we give Voronovskaja theorem for these operators.

Gaussian and Golden Wavelets: A Comparative Study and their Applications in Structural Health Monitoring

In this work, a comparative analysis between Gaussian and Golden wavelets is presented. These wavelets are generated by the derivative of specific base functions. In this case, the order of the derivative also indicates the number of vanishing moments of the wavelet. Although these wavelets have a similar waveform, they have several distinct characteristics in time and frequency domains. These distinctions are explored here in the scale space. In order to compare the results provided by these wavelets for a real signal, they are used in the decomposition of a signal inserted in the context of structural health monitoring.