Variational Principle Research Articles

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

AbstractA variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton–Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot’s formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo’s law for heat conduction, and the well-known Lord–Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction.

Weighted topological pressure revisited

Abstract Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9)106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys.43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

Nonlocal modelling of piezoelectric solid with cracks based on peridynamic differential operator

The paper establishes a nonlocal model for piezoelectric solids using the Peridynamic differential operator. The method transforms the displacement-electric potential equation, constitutive equation, and boundary conditions of piezoelectric solids from a local differential form to a nonlocal integral form. The solution for the displacement and electric potential distributions in piezoelectric solids is obtained using variational principles and the Lagrange multiplier method. The correctness and effectiveness of the model are validated through a comparison of the deformation analysis of piezoelectric solid square plates with analytical solutions. Analysis of piezoelectric solid square plates with edge cracks proves the applicability of this model for solving discontinuous problems. Furthermore, the paper explores the suitability of the model in analysing problems involving multiple cracks in piezoelectric solid plates, highlighting its capability to address defective piezoelectric solid problems. This model provides a new perspective for solving problems involving defects under electro-mechanical coupling conditions.

New bandgap analysis method for metamaterial structures using variational principle

This paper presents an efficient bandgap analysis method for metamaterial structures based on variational principle and linear expression technique. This is in view of the challenges in constructing a displacement function that satisfies the Bloch boundary when applying the traditional energy method to analyze the dispersion relations of metamaterial structures. Also, the inclusion of the wave numbers in the displacement function leads to inefficient computation. While there have been methods, such as virtual springs, have been proposed to deal with such problems, issues remain to be resolved. The main idea of the current paper is the solution of linear uncorrelated coefficients by processing the constraints. In this way, the unknown coefficients in the equations of motion can be linearly expressed, thereby allowing the equations to become variational. Both Gaussian elimination and null-space techniques are used to implement the method. The solution procedures of the two methods are detailed and demonstrated by using one-dimensional and two-dimensional metamaterial structures. The accuracy of the method is verified by comparing the results with those from using finite element method. The calculation efficiency is compared with that of the traditional energy method in terms of both the matrix dimension and the number of the points of wave numbers. The results show that the method has a much higher calculation efficiency than that of the traditional energy method. The method proposed in this paper is free of convergence problems in dealing with boundary problems. Furthermore, in the discussion of combined periodic structures, it is shown that the method still has good geometrical applicability in structures containing multiple boundary conditions, showing great potential in solving complex engineering problems.

Stability and nonlinear vibration characteristics of cantilevered fluid-conveying pipe with nonlinear energy sink

Considering the midline non-expansion theory and nonlinear coordinate method, this paper mainly investigates the stability and nonlinear vibration characteristics of cantilevered fluid-conveying pipe with energy sink. Based on Hamilton 's variational principle, the dynamic equation of a cantilevered pipe conveying fluid with nonlinear energy sink (NES) under pulsating flow is established. The system equation is processed by Galerkin method, a novel matrix equations of single-degree-of-freedom and multi-degree-of-freedom coupling system is established. The results show that the physical factors in the flow pipe system have a great influence on the critical flow velocity of the whole system, comparing with the case of adding NES. Increasing the proportion of the mass of the fluid will have a greater impact on the real part frequency, thereby increasing the critical velocity of the fluid-conveying pipe system. Addition of nonlinear energy sink can significantly suppress the amplitude and decrease the unstable vibration of the pipe.

A new buckling model for thin-walled micro-beams based on modified gradient elasticity: coupling effect and size effect

More and more thin-walled micro-beams, including the I-section thin-walled micro-beams, are applied in MEMS. The buckling of a thin-walled micro-beam exhibits some different phenomena from that of a large scale thin-walled beam, such as the size effect, and the coupling effects caused by strain gradient. To investigate the influence of the coupling effects and size effect on the buckling of thin-walled micro-beams, a new buckling model is proposed. Based on modified gradient elasticity (MGE), the governing equations and boundary conditions of the new model are derived from the variational principle. The governing equations and boundary conditions can be simplified to the classical Vlasov's theory, the MGE Bernoulli-Euler beam model, MGE Bernoulli-Euler beam model considering bi-directional flexure respectively. Solved by the feature expansion method, the size effects of the thin-walled micro-beam can be captured by the new model, that is the buckling load increases with internal length scales. Besides the classical flexural-torsional coupling effect, two new coupling effects, such as the higher-order flexure-flexure coupling and the higher-order flexural-torsional coupling, are also captured by the new model. These two new high-order coupling effects of the new model reduce the critical buckling load of thin-walled micro-beams. The higher-order flexural-flexural coupling effect is far greater than the other two coupling effects, and the higher-order flexural-flexural coupling effect is on the same order of magnitude as the effect of MGE (size effect), with the former being negative and the latter positive. The size effect and higher-order coupling effect cannot be ignored and need to be considered carefully for the buckling of thin-walled micro-beams.

Hydrodynamic density functional theory of simple dissipative fluids

In this paper, a statistical physical derivation of thermodynamically consistent fluid mechanical equations is presented for non-isothermal viscous molecular fluids. The coarse-graining process is based on (i) the adiabatic expansion of the one-particle probability density function around local thermodynamic equilibrium, (ii) the assumption of decoupled particle positions and momenta, and (iii) the variational principle. It is shown that there exists a class of free energy functionals for which the conventional thermodynamic formalism can be naturally adopted for non-equilibrium scenarios, and describes entropy monotonic fluid flow in isolated systems. Furthermore, the analysis of the general continuum equations revealed the possibility of a non-local transport mode of energy in highly compressible dense fluids.

Exploring mean field spin models with multiple populations: A theoretical framework

This paper thoroughly investigates the analysis of a modified generalized mean-field Ising model. The thermodynamic limit of free energy is investigated, and both the upper and lower bounds are derived. The lower bound is derived using a standard entropic variational principle, whereas a decoupling method is employed to derive the upper bound. Although the two bounds differ in form, they exhibit the same local maxima. Finally, the impact of a slight external field on the magnetic order is analyzed.

Introduction to mean dimension

The purpose of this survey is to explain basic ideas of mean dimension theory. Mean dimension represents a number of parameters per unit time for describing a dynamical system. It was introduced by Gromov in 1999. His motivation is to propose a new approach to geometric analysis over noncompact manifolds. Rather independently of this original motivation, Lindenstrauss and Weiss found applications of mean dimension to classical problems in topological dynamics. In this survey we start from the viewpoint of Lindenstrauss–Weiss and first review the applications to topological dynamics. Next we explain the relation to geometric analysis, in particular holomorphic curve theory. Mean dimension is connected to rate distortion theory, which is a branch of information theory studying the trade-off between distortion and compression rate in lossy data compression. In the last three sections we review a variational principle between mean dimension and rate distortion function and then discuss its possible application to holomorphic curve theory.

Comparing finite viscoelastic constitutive relations and variational principles in modeling gastrointestinal soft tissue deformation

The mechanical attributes of soft tissues within the gastrointestinal (GI) tract are crucial for the effective operation of the GI system, and alterations in these properties may play a role in motility-related disorders. Various constitutive modeling approaches have been suggested to comprehend the response of soft tissues to diverse loading conditions. Among these, hyperelastic constitutive models based on finite elasticity have gained popularity. However, these models fall short in capturing rate- and time-dependent tissue properties. In contrast, finite viscoelastic models offer a solution to overcome these limitations. Nevertheless, the development of a suitable finite viscoelastic model, coupled with a variational formulation for efficient finite element (FE) implementation, remains an ongoing challenge. This study aims to address this gap by developing diverse finite viscoelastic constitutive relations and applying them to characterize soft tissue. Furthermore, the research explores the creation of compressible, nearly incompressible, and incompressible versions of viscoelastic constitutive relations, along with their variational formulation, to facilitate efficient FE implementation. The proposed model demonstrates remarkable accuracy in replicating experimental results, achieving an R2 value exceeding 0.99.

Variational Principle for a Fractal Lubrication Problem

Variational Principle for a Fractal Lubrication Problem

The Duality Theory of Fractional Calculus and a New Fractional Calculus of Variations Involving Left Operators Only

Through duality, it is possible to transform left fractional operators into right fractional operators and vice versa. In contrast to existing literature, we establish integration by parts formulas that exclusively involve either left or right operators. The emergence of these novel fractional integration by parts formulas inspires the introduction of a new calculus of variations, where only one type of fractional derivative (left or right) is present. This applies to both the problem formulation and the corresponding necessary optimality conditions. As a practical application, we present a new Lagrangian that relies solely on left-hand side fractional derivatives. The fractional variational principle derived from this Lagrangian leads us to the equation of motion for a dissipative/damped system.

Effect of spatial setting angle on vibration of elastically restrained rotating beams

For a fully articulated rotor system, there may exist a spatial setting angle and elastically restrained root for each individual blade in rotation procedure due to flapping, drag and pitch hinges. Most of previous studies on rotating blades or beams focused on pitch angle and clamped root. In this paper, a three-dimensional dynamic model is proposed for elastically restrained rotating beams with spatial setting angle. The elastic deformation of the beam is expressed by three scalar variables in three-dimensional space including stretch deformation instead of the conventional axial deformation. The longitudinal shrinkage is considered to capture centrifugal forces. The penalty method is adopted to deal with the elastic constraints, which greatly simplifies the construction of admissible functions. A unified formulation is derived by a variational principle combined with Chebyshev orthogonal polynomials. One of the advantages of the present method is that any linearly independent and complete basis functions can be used as admissible function regardless of boundary conditions, and another advantage is that the present method is capable of handling arbitrary boundary conditions by only adjusting the values of penalty factor. The convergence and accuracy of the present method are validated by comparison, and the effects of boundary conditions, angular velocity and spatial setting angle on vibration characteristics are discussed in detail. The results show that natural frequencies increase as the spring stiffness rises in a certain range and the spatial setting angle plays a more significant role on dynamic characteristics for rotating beams.

Variational approach for time-space fractal Bogoyavlenskii equation

In this paper, we apply the variational approach for solving the time-space fractal Bogoyavlenskii equation. By the two-scale fractal complex transformation, the fractal modification of Bogoyarlenskii equation is equivalently rewritten as the original Bogoyavlenskii equation. By variational principle, different type solutions including fractal bright soliton solution, fractal bright-like soliton solution, fractal kinky-bright soliton solution and fractal periodic wave solution are provided, which are not touched in the existing literature. Numerical results are presented to show the propagation behaviours.

Fixed energy solutions to the Euler-Lagrange equations of an indefinite Lagrangian with affine Noether charge

AbstractWe consider an autonomous, indefinite Lagrangian L admitting an infinitesimal symmetry K whose associated Noether charge is linear in each tangent space. Our focus lies in investigating solutions to the Euler-Lagrange equations having fixed energy and that connect a given point p to a flow line $$\gamma =\gamma (t)$$ γ = γ ( t ) of K that does not cross p. By utilizing the invariance of L under the flow of K, we simplify the problem into a two-point boundary problem. Consequently, we derive an equation that involves the differential of the “arrival time” t, seen as a functional on the infinite dimensional manifold of connecting paths satisfying the semi-holonomic constraint defined by the Noether charge. When L is positively homogeneous of degree 2 in the velocities, the resulting equation establishes a variational principle that extends the Fermat’s principle in a stationary spacetime. Furthermore, we also analyze the scenario where the Noether charge is affine.

Optimal control of a multi-scale immuno-influenza A transmission model with viral load-dependent infection

Influenza A is a highly contagious respiratory illness that spreads globally and results in millions of cases worldwide. The transmission rate and mortality rate of influenza in the population are determined by the load of the influenza A virus. In this study, we have developed a multi-scale immuno-influenza model considering incomplete vaccine immunity. We have calculated the basic reproduction numbers for both immunological and epidemiological models. Additionally, we have examined the stability of the disease-free equilibrium and the existence of the endemic equilibrium at the population level. To assess the cost-effectiveness of antiviral therapy at an individual level and vaccination at a population level, we have formulated an optimal control problem. The objective of this problem is to minimize both the number of influenza cases and the cost of vaccination. To solve this optimal control problem, we have employed the Ekeland variational principle. Furthermore, we have designed a global forward and backward sweep algorithm to estimate the impact of antiviral treatment on reducing influenza transmission among humans. Our findings indicate that the use of antiviral treatment significantly reduces viral loads in individuals. However, it may also increase the potential risk at the population level. Therefore, vaccination against influenza is the most effective option from a public health perspective.

Optimal boundary control for a competitive population system with size structure and time delay in a polluted environment

In this paper, we consider the optimal boundary control problem of a two-species competitive system with time delay and size structure in a polluted environment. First, the well posedness of the system is studied by using the characteristic line method and the fixed point principle. Second, the necessary conditions for optimal boundary control are obtained by conjugate system and normal cone property. Finally, the existence and uniqueness of the optimal strategy are proved by Ekeland variational principle.

Vibration transmission model of buildings based on virtual spring model and energy functional variation principle

In recent years, the “metro depot + over-track buildings” development model has witnessed rapid growth in China, driven by its advantages of high land utilization and investment return rate. However, the issue of vibration and noise caused by trains entering and exiting the depot has become a major impediment to its sustainable development. To predict the vibration of over-track buildings induced by trains, this paper presented a vibration transmission model of buildings based on the virtual spring model and the energy functional variation principle. The model was then applied to an actual engineering project at a metro depot in Guangzhou, China, and its accuracy was validated by comparing the results with finite element analysis and field measurements. Additionally, the vibration transmission characteristics of the over-track building were studied. The results indicated that the vibration transmission model of buildings presented in this paper aligns well with the results obtained through finite element analysis and field measurements. This approach eliminates the need to disassemble structural members, and the introduced virtual spring model effectively simulates non-ideal connections and boundary conditions. Consequently, it offers convenient structural connection and boundary condition settings, as well as high computational efficiency. Due to the edge effects on the top floor of the buildings, vibration waves in the higher floors experience a combination of reflected and incident waves, resulting in slow attenuation and, in some cases, amplification of vibrations as they propagate from the middle floors to the top floor. The vibration of the over-track building is closely related to the natural vibration characteristics of the floor and the frequency spectrum characteristics of the vibration source. When the natural frequency of a building's floor aligns with the main frequency of the vibration source, significant vibration amplification occurs due to local resonance.

Combination resonance of a moving ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field

The combination resonance of a moving rectangular ferromagnetic thin plate under double alternating line loads in a transverse constant magnetic field is investigated. Initially, the kinetic and potential energies of the system, considering geometrical nonlinearity, are obtained based on Kirchhoff's theory of thin plates. Additionally, the works of external forces are determined using the principle of virtual work. Subsequently, taking into account the nonlinear magnetization phenomenon of ferromagnetic materials, the magnetization force and Lorentz force acting on the moving ferromagnetic plate are calculated according to electromagnetic theory. Finally, the nonlinear magnetoelastic vibration equation is established by applying the Hamiltonian variational principle. The algebraic equation for static deflection and the differential equation for disturbed deflection are derived by applying the Galerkin method to the vibration equation. Then, an analytical solution for the frequency response equation under combination resonance is obtained using the multiple scale method. The proposed model and analytical research methods are validated by comparison with existing literature and numerical methods. Furthermore, the stability of steady-state solutions is determined, and static bifurcation is analyzed through singularity analysis. With the analytical results, numerical examples are plotted with varying frequency tuning values, magnetic field intensities, load amplitudes and load positions. The effects of these parameters on vibration characteristics are examined by comparing amplitude curves of systems with different physical parameters. The results demonstrate significant nonlinear vibration characteristics of the thin plate under the influence of a constant magnetic field and motion effects.

The Mechanics of Elastomeric Sheet Reinforced With Bidirectional Fiber Mesh Subjected to Flexure on Boundaries

Abstract We investigate the concurrent three-dimensional deformations of fiber-reinforced composite sheets subjected to out-of-plane bending moments via a continuum model, where we invoke the neo-Hookean strain energy model for the matrix material of fiber-reinforced composite, and assimilate the strain energy of fiber reinforcements into the matrix material model by accounting for stretching, bending, and twisting kinematics of the fibers through the computations of the first-order and second-order gradient of deformation. Emphasis is placed on deriving the Euler equation and boundary conditions of bending moment within the framework of the variational principle and configuring composite surfaces using differential geometry. Significant attention has been given to illustrating the concurrent three-dimensional deformation of fiber composite, meshwork deformation, and fiber kinematics. The simulation results reveal that for a square fiber composite subjected to the out-of-plane bending moment, the maximum in-plane deformation of matrix material occurs along the diagonal direction of the domain while the center of the domain experiences weak in-plane deformation. Notably, the matrix material performs isotropic/anisotropic properties depending on the domain size/shape. In addition, the simulated unit fiber deformations reasonably validate the overall deformation of the network, underscoring that the deformations of the embedded fiber units govern the overall mechanical performance of the fiber meshwork. More importantly, the continuum model qualitatively provides reasonable predictions on the damage patterns of construction materials by demonstrating the kinematics of matrix material and meshwork deformation.