Solving Equilibrium Equations Research Articles

Synergistic integration of deep neural networks and finite element method with applications of nonlinear large deformation biomechanics

Patient-specific finite element analysis (FEA) holds great promise in advancing the prognosis of cardiovascular diseases by providing detailed biomechanical insights such as high-fidelity stress and deformation on a patient-specific basis. Albeit feasible, FEA that incorporates three-dimensional, complex patient-specific geometry can be time-consuming and unsuitable for time-sensitive clinical applications. To mitigate this challenge, machine learning (ML) models, e.g., deep neural networks (DNNs), have been increasingly utilized as potential alternatives to finite element method (FEM) for biomechanical analysis. So far, efforts have been made in two main directions: (1) learning the input-to-output mapping of traditional FEM solvers and replacing FEM with data-driven ML surrogate models; (2) solving equilibrium equations using physics-informed loss functions of neural networks. While these two existing strategies have shown improved performance in terms of speed or scalability, ML models have not yet provided practical advantages over traditional FEM due to generalization issues. This has led us to the question: instead of abandoning or replacing the traditional FEM framework that can reliably solve biomechanical problems, can we integrate FEM and DNNs to enhance performance?In this study, we propose a synergistic integration of DNNs and FEM to overcome their individual limitations. Using biomechanical analysis of the human aorta as the test bed, we demonstrated two novel integrative strategies in forward and inverse problems. For the forward problem, we developed DNNs with state-of-the-art architectures to predict a nodal displacement field, and this initial DNN solution was then updated by a FEM-based refinement process, yielding a fast and accurate computing framework. For the inverse problem of heterogeneous material parameter identification, our method employs DNN as a regularizer of the spatial distribution of material parameters, aiding the optimizer in locating the optimal solution. In our demonstrative examples, despite that the DNN-only forward models yielded small displacement errors in most test cases; stress errors were considerably large, and for some test cases, the peak stress errors were greater than 50%. Our DNN-FEM integration eliminated these non-negligible errors in DNN-only models and was magnitudes faster than the FEM-only approach. Additionally, compared to FEM-only inverse method with errors greater than 50%, our DNN-FEM inverse approach significantly improved the parameter identification accuracy and reduced the errors to less than 1%.

Algorithms analysis for solving geometrically nonlinear mechanics problems in the scheme of the semi-analytical finite element method

The effectiveness of using the semi-analytical finite element method (SAFEM) to research geometrically nonlinear construction mechanics problems for 3D bodies of revolution under the arbitrary loads based on a basic ring finite element is considered. An estimate of the rational application parameters of algorithms for taking into account the geometric nonlinearity of a defined above structures class, which has been obtained.
 In the process of numerically solving spatial problems of the theory of elasticity and plasticity with finite displacements, the choice of rational algorithms for solving systems of nonlinear equations is of fundamental importance. It is conditioned by the need of determining the coordinates of the discrete model in the actual configuration and changing the metric characteristics of the finite elements, which, in its turn, leads to the necessity for multiple solutions of systems of nonlinear equations of high order. Due to the introduction of additional hypotheses that do not reduce the accuracy of the approximation: the representation of deformations and stresses in physical terms and in accordance with the moment scheme of the finite element (MSFEM), it is possible, on the one hand, to avoid the time-consuming procedure of numerical integration over the cross-sectional area of the finite element, on the other hand- to maintain a high efficiency of spatial discretization. An important stage in the implementation of computer systems for solving spatial problems is the selection of optimal, from the point of the solution convergence speed and algorithms for solving equilibrium equations. The specificity of the algebraic equations of the SAFEM is conditioned by violation of the trigonometric function orthogonality in the space of the elasticity operator for bodies with variable stiffness and mass parameters along the guide. The clearly defined block structure of the stiffness matrix became the basis for using algorithms combining direct and iterative methods of solving. The reliability of the obtained results and the effectiveness of the approach are confirmed by the solution of a wide range of control examples under various boundary conditions and external loads.

Improvement of the Graphical Method for Plotting the Shear and Moment Diagrams for Members Subjected to Linearly Varying Loads

This study presents an improvement of the graphical method for plotting the shear and moment diagrams for the structural members under linearly varying loads (triangular and trapezoidal loads). Based on the parabolic nature of the shear function, when the loading varies linearly, and on the relations among load, shear, and moment, a mathematical equation is developed to locate the zero-shear point, while a geometric technique is presented to calculate the parabolic shear area. Five comprehensive examples of beams loaded with linearly varying loads are selected to illustrate the steps of the solution for the proposed techniques. The results demonstrated the applicability of the presented method, and gave exact diagrams compared with the basic graphical method. It is concluded that the proposed improved method is generally more convenient, less time-consuming, and has less computational efforts because it does not require sectioning, solving equilibrium equations, and quadratic formulas compared with the basic graphical method.

Performance analysis of a solar still using an absorber plate with inclined perforated rectangular fins: Comprehensive study

Solar desalination are one of the most promising ways to solve the problem of water shortage in the regions that are adjacent to the sea or saltwater bodies that have abundant solar radiation and lack of drinking water. The present research aimed to investigate an experimental and theoretical study to improve the thermal performance of conventional double-slope solar stills by using inclined perforated rectangular fins combined with the distilled basin. Two solar stills are manufactured, the first one is a conventional still, and the other is a finned still. Both stills have a (0.75 m2) cross-section area which is the same area of the distillation basin. Stills were examined during February, March, April, and May under the climatic conditions of Baghdad city in Iraq. The theoretical model relied on solving equilibrium equations of heat and mass with the help of the MATLAB program. The results showed that combining the inclined fins with the distillation basin improves productivity by (33.2%) compared to the conventional solar still. The average productivity of water between February and May is about (3.02-3.6 l/day) and (4.13-4.72 l/day) for the conventional and finned solar still, respectively. The theoretical model was validated using experimental results. The theoretical model predicted well the performance and behavior of the solar still with some deviations from the practical results.

Polynomial functions for configurational correlation functions in Gibbs energies of solid solutions using cluster variation method

The cluster expansion – cluster variation methods (CE – CVM) offer a reliable framework to obtain short range order information for the chosen configurational state of a system accurate up to the basic cluster chosen in terms of the microscopic state variables called correlation functions (CFs). The calculation of the CFs involves solving a set of nonlinear equilibrium equations using compute intensive numerical methods. In this paper a method is presented to approximate the equilibrium values of CFs as polynomial functions of composition which eliminate the necessity of solving equilibrium equations. The coefficients of these polynomials are expressed as rational functions of temperature and system specific energy parameters. This method is comparable to standard CALPHAD methods in its ease of application without sacrificing information about short range order in the system.This procedure is applied to an A2–B32 system using tetrahedron approximation for the case of exclusive second neighbour pair interactions. These polynomials yield CFs comparable to those obtained from numerical solutions in the usual ranges of energy parameters and temperatures.

Dynamic Analysis of Three Snake Robot Gaits

In the present paper, a dynamic analysis is presented, comparing three snake-like robot gaits: lateral undulation, sidewinding locomotion, and sinus-lifting motion. To simplify calculations, sidewinding locomotion and sinus-lifting motion are considered planar movements. Vertical movements are assumed to be small but play a critical role in change where contacts are made. Thus, the normal forces acting on grounded links and the torques applied to pitch joints can be calculated by solving equilibrium equations. The tradeoff between locomotion speed and energy efficiency is studied for all three gaits, at eight different environmental settings distinguished by friction coefficients. Simulation results reveal that sinus-lifting motion and sidewinding locomotion are generally more energy-efficient gaits than is lateral undulation. More specifically, if the anisotropy in friction is large enough, sinus-lifting motion is the most energy-efficient gait; otherwise, sidewinding locomotion is more efficient. However, there are some critical speeds at which the most efficient gait changes, in some environmental settings.

A Comparison of Fourier Spectral Iterative Perturbation Method and Finite Element Method in Solving Phase-Field Equilibrium Equations

AbstractThis paper systematically compares the numerical implementation and computational cost between the Fourier spectral iterative perturbation method (FSIPM) and the finite element method (FEM) in solving partial differential equilibrium equations with inhomogeneous material coefficients and eigen-fields (e.g., stress-free strain and spontaneous electric polarization) involved in phase-field models. Four benchmark numerical examples, including inhomogeneous elastic, electrostatic, and steady-state heat conduction problems demonstrate that (1) the FSIPM rigorously requires uniform hexahedral (3D) and quadrilateral (2D) mesh and periodic boundary conditions for numerical implementation while the FEM permits arbitrary mesh and boundary conditions; (2) the FSIPM solutions are comparable to their FEM counterparts, and both of them agree with the analytic solutions, (3) the FSIPM is much faster in solving equilibrium equations than the FEM to achieve the accurate solutions, thus exhibiting a greater potential for large-scale 3D computations.

Reciprocal diagrams: Innovative applications of past theories

The geometric relationships between a structural configuration and its internal force distribution have not received much attention for nearly hundred years. Directly stemming from the most fundamental principles of statics, the underlying theories were brought to life by the most influential researchers of 19th-century civil engineering such as J.W.M. Rankine, J.C. Maxwell, L. Cremona, and C. Culmann. The graphical approach they proposed for the analysis of trusses soon became a favourite among engineers, offering a less error-prone and faster alternative to solving equilibrium equations by hand. Its popularity faded with the advent of portable calculators and later with the spreading of modern computers. Today, thanks to advances in computer-aided modelling and computational methods, new research tends to reverse the trend. The recent revival, however, has other goals and challenges in mind. Indeed, recent computerised uses of graphical methods are not meant to compete with structural analysis tools. Instead, they aim at opening new horizons for ‘shaping structures’ – that is, the geometric exploration of structural arrangements in static equilibrium – a shift crystallised by Allen and Zalewski’s eponymous book in 1997. More precisely, three main reasons articulate the new direction of research on reciprocal diagrams: (1) they provide an intimate understanding of the intrinsic geometric properties of static equilibrium and therefore its optimisation; (2) their automated construction in a computer-aided environment facilitates interactivity and therefore design opportunities, not only in the form but also the force domain; and (3) the limits of their fundamental principles, long confined to the plane, or at least hardly effective for spatial problems, are now overcome thanks to three-dimensional (3D) computeraided design (CAD) software. The eleven articles in this issue cover the wide spectrum of these three incentives. The first four articles display the elegant theoretical simplifications that occur when dealing with reciprocal diagrams. Mitchell et al. and McRobie et al. explore the connections between mechanisms and states of self-stress and how this can be related to properties of form and force diagrams. Mazurek et al. use reciprocal diagrams for topology optimisation. Williams and McRobie show how discontinuous stress functions provide a direct relationship between bending moments and reciprocal diagrams. The next articles emphasise the role of graphical computation when exploring structural shapes interactively. Fivet reviews the limits of projective transformations. Lee et al. propose new grammar rules for the automated construction of static equilibrium. Zastavni et al. approach the question of robustness at the early design stage. Ohlbrock and Schwartz develop a third diagram to control the topology of the network more explicitly. Fantin and Ciblac extend the solution space of thrust networks for masonry vaults. McRobie develops 3D reciprocal diagrams by means of Minkowski sums. And, finally, Akbarzadeh et al. provide construction methods for the design of spatial structures using reciprocal polyhedra. Early versions of most articles originate from contributions presented at the Symposium of the International Association of Shell and Spatial Structures (IASS) in Amsterdam, 2015, or from the conversations that took place during the special sessions organised at the event by the guest-editors of this issue. We would like to thank all contributors for their valued articles and the many reviewers for the fruitful scientific discussion that has emerged throughout the process. Special thanks go to Noelle Paulson for her assistance with the editorial process. Finally, we are grateful to the editors of the International Journal of Space Structures for the opportunity to share our field of research, gathering some of the most important recent advances, through this journal.

Magnetized neutron stars with superconducting cores: effect of entrainment

We construct equilibrium configurations of magnetized, two-fluid neutron stars using an iterative numerical method. Working in Newtonian framework we assume that the neutron star has two regions: the core, which is modelled as a two-component fluid consisting of type-II superconducting protons and superfluid neutrons, and the crust, a region composed of normal matter. Taking a new step towards more complete equilibrium models, we include the effect of entrainment, which implies that a magnetic force acts on neutrons, too. We consider purely poloidal field cases and present improvements to an earlier numerical scheme for solving equilibrium equations, by introducing new convergence criteria. We find that entrainment results in qualitative differences in the structure of field lines along the magnetic axis.

Necking Prediction in Tube Hydroforming by Stress-Based Forming Limit Diagrams

Path-independent stress based forming limit diagrams for anisotropic materials along with the Hill 48 yield function; power law strain hardening and isotropic hardening are derived. Bifurcation analyses via solving equilibrium equations of a pointed vertex on subsequent yield loci are utilized. The presented practical forming limit diagrams (FLSDs) show more acceptable agreements with experimental data compared with those obtained by other methods. A typical neck detection macro in APDL is developed by FEA simulation for a free bulge tube hydroforming to confirm the significant simplicity and straightforwardness of the mentioned FLSD.

実海域性能を考慮した内航船用船速計画システムの開発

We propose a new navigation system for coastal shipping to decrease fuel consumption and CO2 emission. The navigation system, called WAN-CS, optimizes engine revolutions with constraint of the voyage schedule. For coastal shipping precise forecast system is expected because time of the voyage is short and geographical features of the route must be considered. Thus a new weather forecast system was also developed. The system forecasts four times a day though conventional system provides two times a day. The interval of spatial grid is selected in 2 minutes and the time interval is selected in 1 hour. Ship responses were estimated by solving equilibrium equations of longitudinal force, lateral force and yaw moment considering the engine characteristics. In this paper for reducing fuel consumption a simulation of a coastal cement carrier, which navigated the Ube-Nagoya route using WAN-CS was carried out. It is found that the fuel reduction by WAN-CS achieved 33% from the result of the simulation.

Solving equilibria problems with a graphing calculator: A robust method, free of algebra and calculus

The method of successive approximations is frequently introduced in freshman chemistry as a means for solving equilibrium equations. However, this often requires an understanding of calculus that is problematic for introductory chemistry students. An alternative method to solving such equations is to use graphic calculators to solve problems graphically.

An iterative substructuring algorithm for equilibrium equations

The topic of iterative substructuring methods, and more generally domain decomposition methods, has been extensively studied over the past few years, and the topic is well advanced with respect to first and second order elliptic problems. However, relatively little work has been done on more general constrained least squares problems (or equivalent formulations) involving equilibrium equations such as those arising, for example, in realistic structural analysis applications. The potential is good for effective use of iterative algorithms on these problems, but such methods are still far from being competitive with direct methods in industrial codes. The purpose of this paper is to investigate an order reducing, preconditioned conjugate gradient method proposed by Barlow, Nichols and Plemmons for solving problems of this type. The relationships between this method and nullspace methods, such as the force method for structures and the dual variable method for fluids, are examined. Convergence properties are discussed in relation to recent optimality results for Varga's theory ofp-cyclic SOR. We suggest a mixed approach for solving equilibrium equations, consisting of both direct reduction in the substructures and the conjugate gradient iterative algorithm to complete the computations.