Static Problem Research Articles

Smooth hyperelastic potentials for bimodular materials: 3D case

A family of one parametric infinitely differentiable hyperelastic potentials for three-dimensional problems of bimodular isotropic materials at infinitesimal strain is constructed, yielding a set of uniform approximations to the discontinuous stepwise elastic modulus adopted in the original one-dimensional bimodular formulation. The introduced potentials enable either analytical solutions or construction of the explicit governing equations for a number of static and dynamic problems. Theorem of convergence to the discontinuous bimodular modulus is proved.

Some Applications of the Principle of Virtual Displacements in Physics and Mathematics Courses

Key words: problem, force, statics, work, analogy, inequality, proof The work is devoted to the school application of the principle of virtual displacements. The latter is significant since it is related to the importance of energy approaches in physics with which this principle is associated. Only in university mechanics courses can the idea and its many applications be rigorously and completely proven. However, if the essence of the principle is simply interpreted, then even in the confines of a school physics course it can be effectively used in the process of solving statics problems. It should be noted that it can be challenging to answer some statics problems by taking standard equilibrium conditions of forces and torques into account. In such cases, the only viable method for the solution of these problems is to apply the virtual displacements idea. Although this principle may have interesting applications, it has not been widely used in physics teaching yet. The work's scientific and methodological novelty lies in its clear presentation of the principle of virtual displacements within the context of school physics. These concepts are then applied to the solution of mathematical inequalities as well as statics problems. Using a straightforward example, the article first explains the fundamentals of the idea of the virtual displacement, then examins a sequence of progressively intricate statics. In the article's last section, with the help of some analogies, the concepts of the principle are applied to solve mathematical inequalities.

The link between MiFID and Risk Appetite Framework as an application of best practices for wealth management and the entire value chain of the financial industry

After a short review of the MiFID regulations and the RAF, the paper identifies the link between them which allows to mitigate a balance sheet risk sustained by the financial intermediary and, at the same time, to improve its stability and value creation, through a maximization of customer loyalty. The client’s attitude towards risk can be summarized in these terms: "I don't like risk, but I like to win". Thus, a three-dimensional approach towards expected utility is suggested for estimating risk tolerance: risk aversion, loss aversion and reflection. In addition, a definition of the client's financial objectives is required, combined with greater disclosure - which allows the construction of a financial statement – to attest that risk-taking is indeed a luxury, as indicated by the metrics of the discretionary wealth ratio, and therefore, of the Standard of Living Risk (SLR). The next step consists in the determination of a set of portfolios along the efficient frontier where risk is represented by the expected shortfall, the determination of which belongs to a Generalized Extreme Value Theory logic. The client's objectives are described in terms of probability of success, where the latter is a function of an initial endowment, a potential positive contribution of financial resources over time as well as an expected return level. The above is expressed through a practical case that envisages the determination of a set of EGPF portfolios and the identification of the specific portfolio, obtained as a solution to a static and dynamic optimization problem, where the objectives have been formalized through the calculation of the associated utility

Method of solution composition in contact problems with friction of deformable bodies

A new iterative method for solving static contact problems of two deformable bodies is proposed. The method is based on alternately solving the unilateral contact problem for the first body and the linear elasticity problem with natural boundary conditions for the second body. Fulfillment of Coulomb's friction law involves correction of tangential nodal forces in the sliding area and setting kinematic boundary conditions in the sticking area for the contact boundary of the first body. The goal of solving the linear elasticity problem for the second body is to gradually equalize contact loads on the interacting surfaces. The advantages of the method are demonstrated by solving a number of model examples, including unilateral contact of a linear-elastic plate with a solid foundation, bilateral contact of pressing a deformable block into the foundation, the Hertz problem of contact of two deformable cylinders etc. The method can solve problems on flat and curvilinear contact boundaries.

Modeling quasi-static crack propagation using preconditioned numerical manifold method

The numerical manifold method (NMM) is robust and effective in modeling complex fracture problems as it allows the fracture paths to intersect meshes and uses enrichments with Williams expansions to capture the tip singularities. However, mesh cutting and crack-tip enrichment lead to small cut ratios and linear dependence, respectively, for complicated fracture problems. These two situations constantly cause serious ill-conditioning issues. Therefore, ill-conditioning is extremely challenging for crack propagation modeling. A preconditioning strategy was developed previously for NMM to deal with the ill-conditioning for static crack problems (Liu et al., 2023). In this study, we employ the preconditioned numerical manifold method (PNMM) for complicated crack propagation problems. Furthermore, we use a load factor to ensure that the mechanical mechanism and fracture criterion are satisfied simultaneously for multiple crack propagation problems. Numerical examples demonstrate the remarkable stability and efficiency of PNMM for multiple crack propagation problems.

ANALISIS TEGANGAN STATIK DAN DEFORMASI FRAME ELECTRIC GANESHA SCOOTER PORTABLE (E-GASPOL) MENGGUNAKAN SOFTWARE SOLIDWORKS

This study aims to determine and reduce static stress and increase the safety factor on the E-GASPOL vehicle frame. The type of development research used is the R2D2 type (Reflective, Recursive, Design and Development) which has three phases, namely defining, design and development and dissemination. The data analysis technique used is the finite element method, this method can solve static, dynamic, linear and non-linear problems. In this study, the static stress analysis of vehicle frame was carried out in conditions without rider load (the mass of the frame itself is 13.34 kg) and conditions with rider load (frame mass of 13.34 kg + average rider mass of 70 kg, so the total is 83.34 kg). The static stress analysis results on the four designs, the results of the modified vehicle frame design 3 have the best value, namely obtaining a maximum stress value of 1,255xN/m conditions without rider load, this result gets a decrease of 58.55% from the standard design and with a rider load of 7,699xN/m, this result gets a decrease of 58.56% from the standard design, the smaller the stress value obtained on a frame. The maximum displacement/deformation value of 1,453xmm conditions without rider load, this result gets a decrease of 77.718% from the standard design and with a rider load of 8,791xmm, this result gets a decrease of 78.10% from the standard design. From the data it can be concluded that the modified frame is better and stronger than the standard frame.

Code verification of immersed boundary techniques using the method of manufactured solutions

Code verification plays a crucial role for all finite element applications, especially for non-standard ones, such as immersed boundary approaches, which are typically based on novel algorithms and often error-prone in-house implementations. Instead of relying on rarely available analytical solutions or overkill FEM simulations, in this article, the capabilities of the method of manufactured solutions (MoMS) are explored, enabling an easy and straightforward derivation of closed-form reference solutions. The focus is kept on immersed problems, in particular, on the finite cell method (FCM), and manufactured solutions are derived for 2D and 3D problems involving voids and single/multiple inclusions. We propose several approaches for the construction of the manufactured solutions, where zero traction conditions for void regions and continuous normal stresses along material interfaces are directly fulfilled. Thus, no weak boundary conditions are required for reproducing the manufactured solution via FCM. This not only enables code verification for FCM implementations that lack the option of applying weak boundary conditions, but also keeps the simulation complexity low, when testing other relevant features, e.g., different integration schemes or the implementation of enrichment functions. The flexibility and wide application range of the MoMS in the context of immersed boundary simulations is demonstrated using static, quasi-static, and transient problems in the context of linear elasticity. Finally, the analytical derivations of the manufactured solutions used in this paper are provided as supplementary material.

Theoretical and numerical study on the well-balanced regularized lattice Boltzmann model for two-phase flow.

In the multiphase flow simulations based on the lattice Boltzmann equation(LBE), the spurious velocity near the interface and the inconsistent density properties are frequently observed. In this paper, a well-balanced regularized lattice Boltzmann (WB-RLB) model with Hermite expansion up to third order is developed for two-phase flows. To this end, the equilibrium distribution function and the modified force term proposed by Guo [Phys. Fluids 33, 031709 (2021)1070-663110.1063/5.0041446] are directly introduced into the regularization of the transformed distribution functions when considering the LBE with trapezoidal integral. First, to give a detailed comparison of the well-balanced lattice Boltzmann equation(WB-LBE), WB-RLB, and second-order mixed difference scheme (SOMDS) proposed by Lee and Fischer [Phys. Rev. E 74, 046709 (2006)1539-375510.1103/PhysRevE.74.046709], the theoretical analyses on the force balance of LBE with two different gradient operators, isotropic central scheme (ICS) and SOMDS, as well as the numerical simulations of the stationary droplet are carried out. The force analysis shows that SOMDS can achieve a higher accuracy than ICS for the force balance, which has been validated in the simulations of stationary droplet cases. For the stationary droplet cases, all three models (WB-LBE, WB-RLB, and SOMDS) can capture the physical equilibrium state even at a large density ratio of 1000. Also, the numerical investigations of the WB-RLB model with third-order expansion (WB-RLB3) demonstrate that adjusting the relaxation parameters of the third-order moment can further improve the accuracy and stability of the WB-RLB model. Then, both the droplet coalescence and the phase separation cases are investigated with considering the effect of different interface thickness, which demonstrates that the performance of the WB-RLB for the two-phase dynamic problems is still quite well, and it exhibits better numerical stability when compared with the WB-LBE. In addition, the contact angle problem is investigated by the present WB-RLB model; the numerical results show that the predicted values of the contact angles agree well with the analytical solutions, but the well-balance property is not validated, especially near the three-phase junction. Overall, the present WB-RLB model exhibits excellent numerical accuracy and stability for both static and dynamic interface problems.

Mechanically induced electric potential and charge redistribution in laminated composite piezoelectric semiconductor circular cylindrical thin shells

We study electromechanical effects and piezotronic behaviors in a Kirchhoff–Love laminated composite piezoelectric semiconductor circular cylindrical thin shell, which is composed of a non-piezoelectric semiconductor core and two piezoelectric dielectric layers. A second-order assumption for electric potentials and carrier concentration perturbations is utilized to capture accurate descriptions of the electric field and charge distributions within the composite shell, where the zeroth-order, first-order, and second-order forms represent constant, antisymmetric, and symmetric distributions along the thickness, respectively. The field equations and boundary conditions are simultaneously derived through a principle of virtual work and the fundamental lemma of the calculus of variation. The new governing equations derived show a new coupling relation between mechanical deformation and charge behavior in the composite shell, where all mechanical deformations (extension, bending of the shell) and all distribution forms of electric potential (zeroth-, first-, second-order electric potentials) are coupled, which is different from piezoelectric semiconductor beams and plates. Then, the numerical results of electromechanical effects and piezotronic behaviors in static bending and forced vibration problems of the shell are conducted with the derived theoretical model. Numerical studies of static bending graphically show some fundamental coupling relations between mechanical forces and charge distributions in deformed piezoelectric semiconductor shells. The new results from the forced vibration analysis given by the current model show that the deflection amplitude and the electric potential distribution in the thin shell are frequency-dependent and can be adjusted by controlling the excitation frequency. In addition, the semiconducting properties (dependence of mechanical behavior on doping level) are studied in both static and dynamic problems. The mechanically controlled charge distribution suggests that shell-shaped piezoelectric semiconductor shells could serve as an effective means of sensing or energy harvesting by converting mechanical energy into electricity.

An aspect of the turnpike property. Long time horizon behavior

The turnpike phenomenon concerns the structure of the optimal control and the optimal state of dynamic optimal control problems for long time horizons. The focus is regularly placed on the study of the interior of the time interval. Classical turnpike results state how the solution of the dynamic optimal control problems approaches the solution of the corresponding static optimal control problem in the interior of the time interval.
 In this paper we look at a new aspect of the turnpike phenomenon. We show that for problems without explicit terminal condition, for large time horizons in the last part of the time interval the optimal state approaches a certain limit trajectory that is independent of the terminal time exponentially fast. For large time horizons also the optimal state in the initial part of the time interval approaches exponentially fast a limit state.

Analysis of nonlinear multi-field coupling responses of piezoelectric semiconductor rods via machine learning

ABSTRACT Piezoelectric semiconductors (PSs) have widespread applications in semiconductor devices due to the coexistence of piezoelectricity and semiconducting properties. It is very important to conduct a theoretical analysis of PS structures. However, the present of nonlinearity in the partial differential equations (PDEs) that describe those multi-field coupling mechanical behaviors of PSs poses a significant mathematical challenge when studying these PS structures. In this paper, we present a novel approach based on machine learning for solving multi-field coupling problems in PS structures. A physics-informed neural networks (PINNs) is constructed for predicting the multi-field coupling behaviors of PS rods with extensional deformation. By utilizing the proposed PINNs, we evaluate the multi-field coupling responses of a ZnO rod under static and dynamic axial forces. Numerical results demonstrate that the proposed PINNs exhibit high accuracy in solving both static and dynamic problems associated with PS structures. It provides an effective approach to predicting the nonlinear multi-field coupling phenomena in PS structures.

A hyperbolic theory for the analysis of laminated shallow shells with double curvature

In this paper, higher-order closed-form analytical solutions to the static bending and free vibration problems of laminated composite shells with double curvature are obtained using a hyperbolic shear deformation theory. The current theory is a modification of the shape function provided by Soldatos [30] in his well-known hyperbolic theory. The distributions of transverse shear stresses through the thickness of the shell are precisely predicted by the current theory satisfying traction free boundary conditions at the top and the bottom surfaces of the shell. Hamilton's principle serves as the foundation for the development of equations of motion. Navier's method is used for the analysis of simply-supported laminated shells under static and free vibration conditions. Displacements, stresses, and natural frequencies are presented for different shells with double curvature. The results from past investigations are compared to verify the accuracy and efficacy of the present hyperbolic shell theory.

Ant Metaheuristic Algorithm for the Frequency Assignment Problem

This study proposes the ant metaheuristic algorithm to solve the frequency assignment problem. In this approach, the descent method constructed as three ants which each of which appling a metaheuristics algorithm. Several novel and existing techniques are used to improve the performance of this algorithm. One of these techniques is applying the concept of a well-known graph colouring algorithm, namely recursive largest first. Furthermore, this study compares this algorithm using two visibility definitions. The first definition is based on the number of feasible frequencies and the second one is based on the degree. Additionally, we compare this algorithm using two trail definitions. The first one is between requests and frequencies. The second is between requests and requests. This study considers real and randomly generated benchmark datasets of the static problem and our algorithm achieved competitive results comparing with other ant colony optimization algorithms in the literature.

A Dynamic Ant Colony Optimization for Solving the Static Frequency Assignment Problem

This study proposes a dynamic ant colony optimization algorithm to solve the static frequency assignment problem. This approach solves the static problem by modeling it as a dynamic problem through dividing this static problem into smaller sub-problems, which are then solved in turn in a dynamic process. Several novel and existing techniques are used to improve the performance of this algorithm. One of these techniques is applying the concept of a well-known graph colouring algorithm, namely recursive largest first for each sub-problem. Furthermore, this study compares this algorithm using two visibility definitions. The first definition is based on the number of feasible frequencies and the second one is based on the degree. Additionally, we compare this algorithm using two trail definitions. The first one is between requests and frequencies. The second is between requests and requests. This study considers real and randomly generated benchmark datasets of the static problem and our algorithm achieved competitive results comparing with other ant colony optimization algorithms in the literature.

Dynamic Multiple Neighborhood Structures for the Static Frequency Assignment Problem

This study proposes a dynamic multiple neighborhood structures to solve a variant of the frequency assignment problem known as the minimum order frequency assignment problem. This problem involves assigning frequencies to a set of requests while minimizing interference and the number of used frequencies. Several novel and existing techniques are used to improve the efficiency of this algorithm and make it different from other applications of multiple neighborhood structures in the literature. This includes solving the static problem by modeling it as a dynamic problem through dividing this static problem into smaller sub-problems, which are then solved in turn in a dynamic process using multiple neighborhood structures. Moreover, applying technique that aims to determine a lower bound on the number of frequencies required from each domain for a feasible solution to exist for each sub-problem, based on the underlying graph coloring model. These lower bounds ensure that the search focuses on parts of the solution space that are likely to contain feasible solutions. This study considers real and randomly generated benchmark datasets of the static problem and our approach achieved competitive results.

Nonlinear RNN with noise-immune: A robust and learning-free method for hyperspectral image target detection

While the recurrent neural network (RNN) has achieved remarkable performance on dynamic and control tasks, its applications to image processing, particularly target detection are limited. Challenges arise from differences between the two domains, such as the way for merging time information into static problems and variances of dynamic and static solving methods. To this end, we first extend the existing constrained energy minimization (CEM)-based detection scheme to a dynamic version, e.g. dynamic reinforced CEM (DRCEM), which injects the dynamic information. After that, aided by the rigorous mathematical derivation and optimization theory, the DRCEM is merged into the RNN solution framework. To enhance the robustness and convergence of the existing RNN solutions for improving DRCEM performance, the nonlinear and bounded-constraint RNN (NBCRNN) is designed by developing a novel nonlinear activation function, then applying the proposed model to implement the DRCEM scheme. The corresponding theorem results reveal the proposed model possesses global convergence and enhanced robustness. Compared to state-of-the-art works, the DRCEM solved by the NBCRNN model detection method achieves better detection accuracy, with 1.82% improvement in terms of the Kappa coefficient, and reduces the residual error from 10−4 to 10−7. Furthermore, our detection method is able to preserve the detection accuracy in presence of noise perturbated. To the best of our knowledge, it is the first work to develop the zeroing-type RNN for hyperspectral image target detection. The code and models are publicly available at Github DRCEM_NBCRNN Code Implementation.

Static Problems of Thermoelasticity for a Layered Functionally Graded Thermosensitive Cylinder

Static Problems of Thermoelasticity for a Layered Functionally Graded Thermosensitive Cylinder

Principal Problems of Relativistic Mechanics of Solids

The article deals with a discussion of fundamental problems arising in relativistic mechanics (general relativity) in relation to the determination of stresses generated by gravity in a deformable solid. Three such problems are considered. The first one is associated with the incompleteness of Einstein’s system of equations that includes six mutually independent equations with ten unknown coefficients of the metric tensor. The second one arises when determining the stresses in a solid generated by gravity. For a static problem, three equations of the law of conservation of theory (equilibrium equations) include six unknown stresses, which, unlike Newton’s theory, does not allow determining gravitational stresses. The third problem is related to the reduction of linearized Einstein equations to the equations of Newton’s gravitational theory. Such a reduction turns out to be possible only for empty space and is not valid for a solid body. The noted contradictions can be eliminated by limiting the scope of the theory to a special space, which is Euclidean with respect to spatial coordinates and Riemannian only with respect to time. The discussion is illustrated by a spherically symmetric problem that reduces to ordinary differential equations.

Radio Map-Based Trajectory Design for UAV-Assisted Wireless Energy Transmission Communication Network by Deep Reinforcement Learning

In this paper, we consider a wireless energy-carrying communication network of a UAV. In this communication network, the internet of things (IoT) devices maintain their work via the power supply of batteries. The energy of batteries is slowly consumed over time. The UAV adopts the full-duplex working mode and the flight hover protocol, that is, they can power the target device in the hover position and collect base station data at the same time. Unlike traditional methods, which seek to achieve wireless energy transmission, this paper adopts deep reinforcement learning. On the one hand, the deep reinforcement learning algorithm seeks to solve the dynamic programming problem without a model. The traditional method often requires a prior channel model, to form a formula about variables for convex optimization, while reinforcement learning only requires the interaction between agents and the environment. Then, the strategy is optimized according to the reward function feedback. On the other hand, traditional optimization methods generally solve static programming problems. Since IoT devices constantly collect information from the surrounding physical environment, and their requirements for power supply change dynamically, traditional methods are relatively complex and require huge computational overhead, while deep reinforcement learning performs well in complex problems. The purpose of our work is that with the assistance of the radio map, a UAV can find the best hover position, maximize the energy supplied by the UAV, maximize the data throughput collected, and minimize the energy consumption. The simulation results show that the proposed algorithm can well find the best hovering position of the UAVs and significantly improve the performance of the UAV-assisted wireless energy transmission communication network.

An integrated e-learning approach for teaching statics to undergraduate engineering students

Free body diagrams are a pictorial representation of a body with forces acting on it. Developing the skill to draw and interpret free body diagrams is imperative in an engineering curriculum as it helps students visualize and comprehend the problem. Engineering students tend to have difficulty using free body diagrams due to a lack of spatial understanding, especially when dealing with three-dimensional static problems. Students are often required to draw free body diagram on paper as part of their problem solutions. While this approach is adequate, it still does not allow students to interact with and develop a deeper understanding of free body diagrams. Educational tools developed to build on the understanding of free body diagrams are either limited to a two-dimensional space or does not provide students a comprehensive platform to completely solve a static problem. In this paper, the authors propose a digital platform that allows students to solve both two and three-dimensional free body diagrams for unknown forces. The platform incorporates Merrill’s Principles of Learning and a clear workflow diagram to assist students in better understanding free body diagrams. The proposed solution also presents an improved feedback system to improve the learning experience of students. Furthermore, the proposed solution addresses the needs of instructors by providing them the means to analyze their students’ performance at different stages of the problems.