Adjoint Variational Principle Research Articles

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.

Pulmonary Blood Flow Model Coupled with Breathing Dynamics.

A mathematical model for pulmonary circulation is established taking into account the mechanical behavior coupled with breathing dynamics. The blood flow models in the arterial and venous trees are established by assembling the tube-flow model in each elastic vessel. These models are combined with the sheet-flow model which expresses the blood flow in the capillary network. The finite-element discretization based on adjoint variational principle is applied to solve the model numerically. First, the pulmonary blood flow under a constant gas pressure in the lung is measured using rabbit lungs, and the decrease in blood flow due to gas pressure is observed. The simulation model is verified by comparing the experimental observations with the simulated results, and the causality among the decrease in blood flow, the capillary collapsing and the pressure distribution is discussed. Next, the pulmonary circulation model is coupled with the breathing model to examine the input impedance in normal breathing and to evaluate the effect of varying the inhaled gas pressure in artificial ventilation. It is shown that the results obtained in the simulation correspond to the impedance characteristics observed in experiments in different species. The simulation of artificial ventilation demonstrates that the inhaled gas pressure affects the blood flow not only in arteries and veins but also in the capillaries. These results indicate that the simulation based on the proposed model can be used to investigate the internal mechanical behavior of the pulmonary circulation taking into account the variation due to individuality of the geometry and material properties.

Stability of a column with a follower load and a load-dependent elastic support

The influence of an elastic support on the stability of an elastic column under a follower load is investigated. The support is pinned and has a rotational spring whose stiffness is either constant or increases as the load is applied. Linear and quadratic stiffening functions are treated. An adjoint variational principle and a generalized Rayleigh-Ritz method are developed for this system. Characteristic curves are examined and the critical load, which is associated with flutter instability, is determined. The effects of the initial spring stiffness and the stiffening rate on the critical load are studied. It is found that an increase in either one of these parameters may lower the critical load, which is unexpected, and that support stiffening may have a significant influence on the vibrations and stability of a nonconservative system.

Breathing-Ventilation Model, and Simulation of High-Frequency Ventilation

An extended model is proposed for lung respiration. This model is made to cope with gas diffusion by convection in the airway network, gas exchange at the alveolus, gas transport by blood flow, and gas flow in the airway network induced by diaphragm motion. The submodel for each phenomenon is combined into a total respiratory system, and it is formulated in the context of discretization based on the adjoint variational principle

Analysis of Polymer Flow in Cylinder of Injection Molding Machine. A Method for Analysis of Pressure and Temperature Distributions.

Flow of a polymer in the plasticizing process of an injection molding machine is analyzed. A polymer in the heat barrel of the machine is regarded as an imaginary non-Newtonian fluid irrespective of melting. Its mathematical model is solved by FEM based on the adjoint variational principle. In order to obtain initial states in the entire analyzed area, the steady-state problem is first solved. Next, using its results as initial values, the original nonsteady problem is analyzed. Moreover, the boundary conditions are also considered.

Breathing and ventilation model, and simulation of high-frequency ventilation.

An extended model is proposed for lung respiration. This model is made to cope with the gas diffusion by convection in the airway network, gas exchange at the alveolus and gas transport by blood flow, as well as the gas flow phenomenon in the airway network induced by diaphragm motion. The submodel for each phenomenon is combined into a total respiratory system, and it is formulated in the context of discretization based on the adjoint variational principle. The model is examined first in terms of normal respiration and then high-frequency ventilation (HFV) is studied. The effects of HFV are evaluated through several case studies and it is quantitatively shown that the influence of the convective diffusion is dominant in the gas transport in the case of HFV. The proposed model is expected to be used to determine proper parameters of HFV.

高速プロペラの振動揚力面理論の解法について

A lifting-surface theory of oscillating high-speed propellers has already been formulated by the author in a form of integral equations. The purpose of this note is to propose a method for solution of these integral equations. An adjoint variational principle, equivalent to the integral equations for direct and reverse flows as well as to the Kutta conditions in both the flows, is derived. It is shown that the generalized aerodynamic forces acting on the propeller blades will be computed advantageously by applying a Rayleigh-Ritz type method to the variational principle.

Shape detemination of a cantilever column subjected to follower force.

This paper deals with the shape determination problem of the cantilever column, i. e. Beck's column, subjected to the follower force. The objective is to maximize the critical load under constraints of volume constancy and lower limit of design variables, under the influence of internal and external damping. The governing equation of motion is transformed into a matrix system by the finite element method with the adjoint variational principle based on the Lagrange multiplier method. The shape improvement process is formulated by the gradient projection method based on the sensitivity of critical load to the design. Numerical results show that improved shape has a higher critical load than the initial shape (uniform) and makes three eigenvalues achieve a critical point simultaneously.

A simulation of respiratory dynamics and the surface tension influence.

The respiratory dynamics of an excised lung are studied in terms of a numerical simulation to determine the surface tension influence. For a model of lung dynamics consisting of an airway network and parenchyma subsystems, the gas-tissue interaction including the surface tension on the parenchymal microstructure is taken into account. Under a given environment and considering the interdependence of the subsystems, the flow equations modified by an airway elasticity of finite deformation are solved by a finite element scheme based on the adjoint variational principle. A simple but reasonable model is proposed and incorporated into the total model to describe the surface tension influence. Though several case studies with this model, it is demonstrated that the surface tension is significantly important to lung dynamics including stability of respiration.

Stability of a rotating pretwisted nonuniform cantilever with a tip mass subjected to dissipative and follower forces

The stability of a rotating pretwisted non-uniform viscoelastic cantilever beam subjected to a tangential force applied at its free end is determined by a method of approximation based on an adjoint variational principle. The coupled equations of motion are derived from a conservation law, the adjoint boundary value problem is introduced, and an approximate stability determinant is developed from the variational principle. The stability determinant is solved numerically for a variety of choices of values for the rotary inertia of the beam, transverse and rotary inertia properties of a mass capping the free end, the rotational speed, and the pretwist angle, and several graphs are presented to show the influence of these parameters upon the value of the critical flutter load.

Conservation laws for materials exhibiting power-law creep

Conservation laws are derived from materials whose constitutive behavior is characterized by power-law creep with elastic strains. This is accomplished by formulating an adjoint variational principle which has as its Euler-Lagrange equations the governing equations as well as a set of adjoint equations involving adjoint variables. Conservation laws are then derived by an application of Noether’s theorem to the variational principle. The results are analogous to those obtained in linear elasticity, in that conservation laws are shown to arise from translations of spatial and temporal coordinates, rigid-body rotations, and self-similar scalings. A path-independent integral formulation of one of the conservation laws, valid under special circumstances, is derived.

Stability of a pretwisted tapered cantilever beam subjected to dissipative and follower forces

Stability of a pretwisted tapered cantilever beam of rectangular cross-section subjected to a follower force at its free end is investigated. The effects of internal and external damping are included in the study. The non-self-adjoint boundary value problem is formulated with the Euler-Bernoulli theory and an associated adjoint boundary value problem is introduced. Approximate values of the critical load are calculated on the basis of a suitable adjoint variational principle for several values of the geometric and material parameters of the beam. The results are shown in graphs.

Influence of an elastic end support on the stability of a nonuniform cantilever subjected to dissipative and nonconservative forces

The influence of an elastic end support on the stability of a damped, linearly tapered cantilever of rectangular cross section subjected to a follower-end-load is studied. The equation of motion is formulated within the Euler-Bernoulli theory for the case of a Kelvin model viscoelastic beam. The effect of external damping is also included in the partial differential equation of motion. The associated adjoint boundary value problem is derived and appropriate adjoint variational principle is introduced. This variational principle is used as the basis for determining approximately the values of the critical load of the system as it depends upon the taper parameters and the stiffness of the elastic end support. It is found that for a given damped, tapered beam the introduction of an elastic support may destabilize the system in certain eases. Furthermore, the critical value of the stiffness of elastic support at which the instability mechanism changes from flutter to divergence or vice versa is decreased as taper increases. For a given support, an increase in taper also decreases the value of the critical load of the system. These reducing effects are more pronounced with increase in thickness taper. The familiar destabilization phenomenon due to small internal damping is observed from the numerical results, but sufficiently large internal damping may have a stabilizing or destabilizing effect depending on the stiffness of the elastic support.

Stability of a nonuniform cantilever subjected to dissipative and nonconservative forces

The stability of a tapered cantilever beam subjected to a circulatory force at its free end is investigated. The effects of internal and external damping are included in the partial differential equation of motion. An adjoint variational principle has been used to determine approximately the values of the critical flutter load of the system. Graphs which demonstrate the variation of the critical nutter load with taper, damping and tangency coefficient are presented.

Effect of angle of attack on the stability of a rotating non-uniform cantilever with a tip mass subjected to dissipative and non-conservative forces

The influence of angle of attack on the stability of a rotating viscoelastic tapered cantilever beam of rectangular cross-section carrying a tip mass and subjected to a circulatory force at its free end is investigated. The effect of external damping is included in the study. The non-self-adjoint boundary value problem is formulated with the Euler-Bernoulli hypothesis and an appropriate adjoint boundary value problem is introduced. Approximate values of the critical flutter load are calculated on the basis of an adjoint variational principle for several values of geometric and material parameters of the system. The results are presented through a series of graphs.

Stability of a non-uniform beam on a viscoelastic foundation with a shear layer subjected to a non-conservative force

The stability of a cantilever beam of linearly variable rectangular cross-section lying on a viscoelastic foundation with a shear layer subjected to a circulatory force at its free end is investigated. The materials of the beam and the foundation are assumed to be Kelvin-type solids. Approximate values of the critical loads are calculated on the basis of an adjoint variational principle. The influence of geometric and material properties of the system on the critical load is depicted through several graphs. It is found that a shear layer may destabilize the system in certain cases.

The influence of tip mass on the stability of a rotating cantilever subjected to dissipative and transverse follower forces

The state of stability of a rotating viscoelastic cantilever beam subjected to a transverse follower force applied at its free end in the plane of rotation is determined by a method of approximation that is based upon an adjoint variational principle. Particular attention is devoted to the determination of the dependence of the critical flutter load of the system on the transverse, twisting, and rotary inertia properties of a mass capping the free end of the beam. The equations of motion are derived from a conservation law, the adjoint boundary value problem is introduced, and an approximate stability determinant is developed from the variational principle upon assuming a set of coordinate functions which satisfy a selected set of boundary conditions. The stability determinant is solved numerically for a variety of choices of values for the internal damping, the hub radius, tip mass inertia, the rotational speed, and warping rigidity parameters, and several graphs are presented to show the influence of these parameters upon the value of the critical flutter load.

Application of an Iterative Ritz Procedure to a Non‐Conservative Stability Problem

AbstractThe possibility of combining the Ritz procedure with an iteration procedure that can be used to solve certain non‐conservative problems of elastic stability is investigated. This procedure, which is based upon the use of an adjoint variational principle with only a pair of coordinate functions that are successively improved by means of an iteration procedure, is applied to an example problem consisting of a cantilevered column subjected to a tangential force at its free end. Numerical results indicate that the iterated coordinate functions lead to significantly improved values for the critical flutter load and flutter frequency.

Mass optimization of non-conservative cantilever beams with internal and external damping

An adjoint variational principle has been developed for a non-conservatively loaded cantilever beam with Kelvin-Voigt internal and linear external damping and is applied to a beam with a linearly distributed tangential load acting along the centerline of the beam. Relative mass optimization for beams of both rectangular and circular crosssections is considered from a graphical standpoint and from the viewpoint of a computer optimization routine with data given and discussed in both instances. In going to a Rosenbrock optimization routine for beams of rectangular cross-section with a minimum tip thickness constraint imposed it was quite clear that mass ratio reductions in the range 14·9 % to 38 % are possible and that the values of internal and external damping appear influential in determining just how much of a mass reduction is possible. Similarly, for beams of circular cross-section a Rosenbrock optimization routine with a minimum tip diameter constraint imposed showed that mass ratio reductions of the order of 27 % are possible.

Stability of a rotating cantilever subjected to dissipative, aerodynamic, and transverse follower forces

The problem of determining the state of stability of a slender cantilever rotating at a uniform speed and subjected to aerodynamic, dissipative, and transverse follower forces is investigated. The equations of motion for the flexural-torsional deformations of a slender rotating cantilever bar are derived from a conservation law that embodies the field equations for the disturbed state of stress in a solid rotating about a fixed axis relative to the initial state of stress associated with the undisturbed form of equilibrium of the rotating system. The associated non-self-adjoint boundary value problem is determined, and a suitable adjoint variational principle is derived. This variational principle serves as the basis for determining approximately the complex eigenvalues and hence the critical flutter values of the transverse follower force for prescribed values of the geometric and material properties of the system. Several graphs which illustrate the variation of the critical flutter load with the speed of rotation, hub radius, the internal damping parameters, aerodynamic load parameters, and the warping rigidity are presented.