Mixed Variational Approach Research Articles

A mixed variational approach for modeling frictional contact problems with normal compliance in electro-elasticity

A mixed variational approach for modeling frictional contact problems with normal compliance in electro-elasticity

Optimal robust online tracking control for space manipulator in task space using off-policy reinforcement learning

This study addresses the demands for adaptability, uncertainty management, and high performance in the control of space manipulators, and the inadequacies in achieving optimal control and handling external uncertainty in task space in previous research. Based on off-policy reinforcement learning, a model-free and time-efficient method for online robust tracking control in task space is devised. To address the complexity of dynamic equations in task space, a mixed-variable approach is adopted to transform the multivariable coupled time-varying problem into a single-variable problem. Subsequently, the optimal control policy is derived with the disturbance convergence, stability, and optimality of the control method being demonstrated. This marks the first instance of achieving robust optimal tracking control in task space for space manipulators. The efficacy and superiority of the presented algorithm are validated through simulation.

Effect of network architecture on physics-informed deep learning of the Reynolds-averaged turbulent flow field around cylinders without training data

Unsupervised physics-informed deep learning can be used to solve computational physics problems by training neural networks to satisfy the underlying equations and boundary conditions without labeled data. Parameters such as network architecture and training method determine the training success. However, the best choice is unknown a priori as it is case specific. Here, we investigated network shapes, sizes, and types for unsupervised physics-informed deep learning of the two-dimensional Reynolds-averaged flow around cylinders. We trained mixed-variable networks and compared them to traditional models. Several network architectures with different shape factors and sizes were evaluated. The models were trained to solve the Reynolds-averaged Navier-Stokes equations incorporating Prandtl’s mixing length turbulence model. No training data were deployed to train the models. The superiority of the mixed-variable approach was confirmed for the investigated high Reynolds number flow. The mixed-variable models were sensitive to the network shape. For the two cylinders, differently deep networks showed superior performance. The best fitting models were able to capture important flow phenomena such as stagnation regions, boundary layers, flow separation, and recirculation. We also encountered difficulties when predicting high Reynolds number flows without training data.

Fully analytical solution framework for general thin-walled composite beams with mixed variational approach

A variationally consistent analytical beam model that describes the theory in a Timoshenko-Vlasov level is developed based on Reissner’s mixed variational theorem. Starting from a shell theory, all the field-governing equations (equilibrium and continuity) and the boundary conditions of the shell wall are derived in closed form, and the mixed method enables finding the explicit forms of the reactive stresses and sectional warpings which are evaluated progressively depending on the level of beam model to be analyzed. The stress recovery part is incorporated in the post-stage of the analysis to compute the layer-wise distribution of stresses over the beam cross-section. The present analysis is validated against numerous benchmark examples available in the literature, including beams with multi-layered strip section, thin-walled anisotropic box sections with elastic couplings, and two-cell airfoil section. The comparison study demonstrates excellent correlations with the results from detailed three-dimensional finite element analysis and other up-to-date beam approaches. Also presented are symbolically expressed stiffness coefficients and the sectional warping modes of coupled composite beams to demonstrate the strength of the proposed beam model.

Non-homogeneous Dirichlet-transmission problems for the anisotropic Stokes and Navier-Stokes systems in Lipschitz domains with transversal interfaces

This paper is build around the stationary anisotropic Stokes and Navier-Stokes systems with an L^infty -tensor coefficient satisfying an ellipticity condition in terms of symmetric matrices in {mathbb {R}}^{ntimes n} with zero matrix traces. We analyze, in L^2-based Sobolev spaces, the non-homogeneous boundary value problems of Dirichlet-transmission type for the anisotropic Stokes and Navier-Stokes systems in a compressible framework in a bounded Lipschitz domain with a transversal Lipschitz interface in {mathbb {R}}^n, nge 2 (n=2,3 for the nonlinear problems). Thus, the interface intersects transversally the boundary of the Lipschitz domain and divides the domain into two Lipschitz sub-domains. First, we use a mixed variational approach to prove the well-posedness of linear problems related to the anisotropic Stokes system. Then we show the existence of a weak solution to the Dirichlet and Dirichlet-transmission problems for the nonlinear anisotropic Navier-Stokes system. This is done by implementing the Leray-Schauder fixed point theorem and using various results and estimates from the linear case, as well as the Leray-Hopf and some other norm inequalities. Explicit conditions for uniqueness of solutions to the nonlinear problems are also provided.

A strain-displacement mixed formulation based on the modified couple stress theory for the flexural behaviour of laminated beams

A novel strain-displacement variational formulation for the flexural behaviour of laminated composite beams is presented, which accurately predicts three-dimensional stresses, yet is computationally more efficient than 3D finite element models. A global third-order and layer-wise zigzag profile is assumed for the axial deformation field to account for the effect of both stress-channelling and stress localisation. The axial and couple stresses are evaluated from the displacement field, while the transverse shear and transverse normal stresses are computed by the interlaminar-continuous equilibrium conditions within the framework of the modified couple stress theory. Then, axial and transverse force equilibrium conditions are imposed via two Lagrange multipliers, which correspond to the axial and transverse displacements. Using this mixed variational approach, both displacements and strains are treated as unknown quantities, resulting in more functional freedom to minimise the total strain energy. The differential quadrature method is used to solve the resulting governing and boundary equations for simply-supported, clamped and cantilever laminated beams. The deflections and stresses from this variational formulation for simply supported beams agree well with those from a Hellinger-Reissner stress-displacement mixed model found in the literature and the 3D elasticity solution given by Pagano. These strain-displacement models also accurately predict the localised stresses near clamped and free boundaries, which is confirmed by the high-fidelity Abaqus models.

On relaxation of some customized proximal point algorithms for convex minimization: from variational inequality perspective

The proximal point algorithm (PPA) is a fundamental method for convex programming. When applying the PPA to solve linearly constrained convex problems, we may prefer to choose an appropriate metric matrix to define the proximal regularization, so that the computational burden of the resulted PPA can be reduced, and sometimes even admit closed form or efficient solutions. This idea results in the so-called customized PPA (also known as preconditioned PPA), and it covers the linearized ALM, the primal-dual hybrid gradient algorithm, ADMM as special cases. Since each customized PPA owes special structures and has popular applications, it is interesting to ask wether we can design a simple relaxation strategy for these algorithms. In this paper we treat these customized PPA algorithms uniformly by a mixed variational inequality approach, and propose a new relaxation strategy for these customized PPA algorithms. Our idea is based on correcting the dual variables individually and does not rely on relaxing the primal variables. This is very different from previous works. From variational inequality perspective, we prove the global convergence and establish a worst-case convergence rate for these relaxed PPA algorithms. Finally, we demonstrate the performance improvements by some numerical results.

A refined two‐dimensional theory for thermal bending of functionally graded plates with thickness normal strains

AbstractA refined theory is presented for the thermal response of composite functionally graded plates under thermo‐mechanical loads. The theory includes shear deformation and normal strain effects by assuming through‐the‐thickness linear in‐plane displacements and parabolic transverse displacement. A mixed variational approach is used to introduce stresses satisfying the loading conditions on the plate surfaces without employing any shear correction factor used in other theories of the same order. A thermal bending problem is solved for metal/ceramic rectangle plates. Some numerical results are compared with numerical results of 3D exact solutions. The importance of the normal strain effect for such problems is studied.

An electromechanically coupled intrinsic, mixed variational formulation for geometrically nonlinear smart composite beam

• Geometrically exact formulation for smart slender structures. • Non-linear analysis of piezoelectric laminated composite beam. • Excellent agreement of results with the literature as well as the finite element software, ABAQUS. • Development of a platform for faster analysis while maintaining the accuracy. The work presented in this article is the outcome of a combined strategy of a mathematical tool for 2D cross-sectional analysis, i.e., Variational Asymptotic Method (VAM) as well as the 1D exact beam analyzer, i.e., the intrinsic mixed variational formulation for modeling and analysis of Piezoelectric-laminated composite beams. This work talks about a novel approach of mixed variational formulation to analyze a two-way electromechanically coupled piezoelectric composite beam. In a classical intrinsic mixed variational approach for a passive structure, the 1D exact beam model deals only with mechanical degrees of freedom. In the present case, an extra 1D electrical degree of freedom has been incorporated. A computational code is developed based on the present theory to solve the two-way coupled electromechanical beam problem. In the present case, we have validated the static results for sensor application. Both linear and nonlinear results have been discussed. Results obtained are very promising and are helpful in building a platform where design, optimization and nonlinear analysis of composite ‘smart’ beams in a multibody framework can be done faster while maintaining acceptable accuracy.

Model-independent determinations of the electron EDM and the role of diamagnetic atoms

We perform model-independent analyses extracting limits for the electric dipole moment of the electron and the P,T-odd scalar-pseudoscalar (S-PS) nucleon-electron coupling from the most recent measurements with atoms and molecules. The analysis using paramagnetic systems, only, is improved substantially by the inclusion of the recent measurement on HfF+ ions, but complicated by the fact that the corresponding constraints are largely aligned, owing to a general relation between the coefficients for the two contributions. Since this same relation does not hold in diamagnetic systems, it is possible to find atoms that provide essentially orthogonal constraints to those from paramagnetic ones. However, the coefficients are suppressed in closed-shell systems and enhancements of P,T-odd effects are only prevalent in the presence of hyperfine interactions. We formulate the hyperfine-induced time-reversal-symmetry breaking S-PS nucleon-electron interaction in general atoms in a mixed perturbative and variational approach, based on electronic Dirac-wavefunctions including the effects of electron correlations. The method is applied to the Hg atom, yielding the first direct calculation of the coefficient of the S-PS nucleon-electron coupling in a diamagnetic system. This results in additionally improved model-independent limits for both the electron EDM and the nucleon-electron coupling from the global fit. Finally we employ this fit to provide indirect limits for several paramagnetic systems under investigation.

Effects of Moment Gradient on Elastic Flange Local Buckling of I-Shaped Beams

Recently, after clearly pointing out the flaws associated with the current AISC equation for flexural strength of I-shaped beams with slender flange, the authors proposed new equation, more accurate and more consistent than the current AISC equation, based on the mixed variational approach. However, the new equation was derived under the condition of uniform moment loading. In this paper, elastic flange local buckling strength formula for I-shaped beams with slender flange under moment gradient was proposed as a follow-up study on the basis of mixed variational approach and supplemental numerical works. The flange plate buckling coefficient predicted by the formula of this study is again shown to be much more accurate and consistent.

Elastic flange local buckling of I-shaped beams considering effect of web restraint

The 2010 AISC flexural strength of I-shaped beams with slender flange was proposed based on Johnson (1985)’s experimental plate bucking coefficient (kc). However, our close examination revealed that most of I-beam specimens tested by Johnson to derive kc had section configuration of slender web and noncompact flange, indicating the inappropriateness of the test database itself and probable underestimation of web restraint effect in the AISC codification. In this paper, the effect of web slenderness on the elastic flange local buckling is analytically investigated using the mixed variational approach, and more accurate but still practical formula is proposed.

A powerful method to analyze of photonic crystals: mixed variational method

Photonic crystal has drawn much attention because of its application in molding the flow of light, which can be used in optical communication, optical storage and computing. In theory, plane wave expansion method, finite difference time domain (FDTD) method and transfer matrix method are widely used methods to study photonic crystal, and each of them has its own advantages and disadvantages.Here, a new method i.e. mixed variational method is introduced to study the photonic crystals, which is from the work of anti-plane shear waves in periodic layered elastic composites. The calculations of this method are direct and require no iteration, which accurately and efficiently produce the entire band structure of the composite and other field characteristics. Moreover, the composite cell in this method may consist of any number of units of any variable permittivity and permeability.Firstly, based on the variational principle, the Lagrangian density of electro-magnetic field is obtained. Then through the surface integral of the Lagrangian density in the unit cell, the Lagrangian is acquired. The first variation of Lagrangian with respect to electric field and magnetic field yields a set of Euler-Lagrange equations. Approximate solutions in explicit series expressions subject to the Bloch periodicity are substituted into the above equations. Minimization of Lagrangian with respect to the electric field and magnetic field results in an eigenvalue problem, and to solve it, the band structure of the composite is yielded. Electrical field, magnetic field, group velocity and energy flux density are also calculated. Secondly, we use the above method to study a two dimensional air-rod unit cell system. Bandgaps with respect to different structural parameters are plotted, which are the same as the results from the plane wave expansion method and FDTD method. In theory, the entire band structure can be calculated with our method. There are more gaps for TE wave than for TM case. By constant frequency contours, it is shown that there is a gap between the first and the second pass band for TE wave, however, there is no a gap for the corresponding TM wave. The directions of group velocity for the first and the second bands are shown in the contours. Electrical field, magnetic field and energy flux in cells illustrate the energy distribution, and the energy-flux directions and the group-velocity directions are also essentially the same. Lastly, we apply this mixed variational method to one-dimensional media-air slab and three dimensional sphere-air structure. The obtained band results accord with those reported previously former, which demonstrates that our method is universal and correct.In the present work, a mixed variational approach is proposed to produce the entire band structure of the composite for unit cells with any arbitrary properties. Explicit expressions are developed for the band, electrical field, magnetic field, group velocity and energy flux.

An exact spectral-dynamic stiffness method for free flexural vibration analysis of orthotropic composite plate assemblies – Part I: Theory

A spectral-dynamic stiffness method (S-DSM) for exact free vibration analysis of general orthotropic composite plate-like structures is presented in the paper. The method combines the advantages of the classical dynamic stiffness method (DSM) with those of the spectral method, and it resembles the finite element and the boundary element methods. The formulation is based on the exact general solution of the governing differential equation, which provides complete flexibility to describe any arbitrary boundary conditions. The dynamic stiffness formulation is essentially accomplished through a mixed-variable approach in a symbolic form with explicit expressions rendering physical meanings. Then a systematic procedure for plate assemblies under arbitrary boundary constraints is described. Finally, a set of novel techniques are proposed to enhance the Wittrick–Williams algorithm by resolving the fully-clamped plate problem. The validation of the theory and its applications to a wide range of engineering structures are demonstrated in Part II of this two-part paper.

Analyses of freely vibrating cross-ply laminated plates in conjunction with adaptive global piecewise-smooth functions (A-GPSFs)

In this paper freely vibrating multilayered plates are analyzed through a suitable adaptive set of global piecewise-smooth functions (or A-GPSFs). Such a subset of functions is herein singled out from a mother set (GPSFs) formerly introduced by Messina [Messina A. Free vibrations of multilayered plates based on a mixed variational approach in conjunction with global piecewise-smooth functions. J. Sound Vib. 2002, 256(1): 103–129.] in 2002; the interesting novelty of this subset consists in the possibility to select the number of functional components for each single layer of the plate by preserving, or checking in a self-contained way, both the accuracy of the 3D-results and the global characteristics of the GPSFs. The governing differential equations and associated boundary conditions are consistently derived from the classical theorem of virtual displacements and are able to treat the multilayered plate as if it were made up of a single layer. The relevant model is essentially two-dimensional along with an expansion through the thickness aimed at modeling a three-dimensional dynamical behavior. The generalization, the simplicity and the accuracy of the present model, based on adaptive global piecewise-smooth functions, result from the comparison of the associated eigen-parameters with those of other models and with those of the exact three-dimensional theory.

Resolving issues with member buckling in truss topology optimization using a mixed variable approach

In this paper, several issues related to member buckling in truss topology optimization are treated. In the conventional formulations, where cross-sectional areas of ground structure members are the design variables, member buckling constraints are known to be very difficult to handle, both numerically and theoretically. Buckling constraints produce a feasible set that is non-connected and non-convex. Furthermore, the so-called jump in the buckling length phenomenon introduces severe difficulties for determining the correct buckling strength of parallel consecutive compression members. These issues are handled in the paper by employing a mixed variable formulation of truss topology optimization problems. In this formulation, member buckling constraints become linear. Parallel consecutive members of the ground structure are identified as chains, and overlapping members are added to the ground structure between each pair of nodes of a chain. Buckling constraints are written for every member, and linear constraints on the binary member existence variables disallow impractical topologies. In the proposed approach, Euler buckling as well as buckling according to various design codes, can be incorporated. Numerical examples demonstrate that the optimum topology depends on whether the buckling constraints are derived from Euler's theory or from design codes.

A mixed discrete-continuous variable multiobjective genetic algorithm for targeted implementation of nonpoint source pollution control practices

[1] Implementation of nonpoint source (NPS) pollution control strategies at the watershed scale hinges on abating pollutant movement from the landscape to water bodies at minimum cost. This paper presents an integrated simulation-optimization approach for targeted implementation of agricultural conservation practices at the watershed scale. A multiobjective genetic algorithm (NSGA-II) with mixed discrete-continuous decision variables was coupled with a distributed watershed model, Soil and Water Assessment Tool (SWAT), to identify optimal types and locations of conservation practices for nutrient and pesticide control at the watershed scale. Previous optimization studies have used binary representation of nonpoint source pollution controls, even though many could be better characterized as continuous variables. In this study, a novel discrete-continuous decision variable, also known as mixed-variable, representation was used to enhance the versatility of the approach by evaluating more options during the search process. Application of the proposed framework in the Eagle Creek Watershed, Indiana, indicated that the optimal suite of conservation practices from the mixed-variable NSGA-II was more effective in meeting water quality targets at lower costs than the solution from binary-variable optimization. However, the mixed-variable approach was considerably more computationally demanding for assessing tradeoffs between environmental and economic factors. A method for hybridization of binary and mixed-variable NSGA-II methods in the context of nonpoint source pollution control practices was developed to enhance the computational efficiency of the optimization procedure. As a result, the number of model simulations required for convergence to the Pareto-optimal solutions was reduced by 96%. The conceptual complexity and computational requirements of optimization-based approaches are impediments to their wider application for targeted implementation of NPS pollution control strategies. The methods and finding of this study address these issues and could result in a more effective implementation of management strategies at the watershed scale.

Variational gradient plasticity at finite strains. Part II: Local–global updates and mixed finite elements for additive plasticity in the logarithmic strain space

The second part of our work on variational inelasticity with long-range effects outlines the formulation and finite element implementation of additive finite gradient plasticity in the logarithmic strain space. It is considered to be the most simple approach to finite plasticity, suitable for the purely phenomenological description of polycrystalline metals or amorphous materials, if structures of the geometrically linear theories are defined in the Lagrangian logarithmic strain space. We start from a mixed saddle point principle for metric-type additive plasticity, which is specified for the important model problem of isochoric von Mises plasticity with gradient-extended hardening/softening response. The mixed variational structure includes the hardening/softening variable itself as well as its dual driving force. The numerical implementation exploits the underlying variational structure, yielding a canonical symmetric structure of the monolithic problem. It results in a novel finite element design of the coupled problem incorporating a long-range hardening/softening parameter and its dual driving force. This allows a straightforward local definition of plastic loading–unloading driven by the long-range fields, providing very robust finite element implementations of gradient plasticity. It includes a rational method for the definition of elastic–plastic-boundaries (EPBs) in gradient plasticity along with a postprocessor that defines the plastic variables in the elastic range. We discuss alternative mixed finite element designs of the coupled problem, including a local–global solution strategy of short-and long-range fields. All methods are derived in a rigorous format from variational principles. Numerical benchmarks demonstrated the excellent performance of the proposed mixed variational approach to gradient plasticity in the logarithmic strain space.

An advanced finite element formulation for piezoelectric beam structures

Since many piezoelectric components are thin rod-like structures, a piezoelectric finite beam element can be utilized to analyse a wide range of piezoelectric devices effectively. The mechanical strains and the electric field are coupled by the constitutive relations. Finite element formulations using lower order functions to interpolate mechanical and electrical fields lead to unbalances within the numerical approximation. As a consequence incorrect computational results occur, especially for bending dominated problems. The present contribution proposes a concept to avoid these errors. Therefore, a mixed multi-field variational approach is introduced. The element employs the Timoshenko beam theory and considers strains throughout the width and the thickness enabling to directly use 3D constitutive relations. By means of several numerical examples it is shown that the element formulation allows to analyse piezoelectric beam structures for all typical load cases without parasitically affected results.

Shape derivatives in differential forms I: an intrinsic perspective

We treat Zolésio’s velocity method of shape calculus using the formalism of differential forms, in particular, the notion of Lie derivative. This provides a unified and elegant approach to computing even higher-order shape derivatives of domain and boundary integrals and avoids the tedious manipulations entailed by classical vector calculus. Hitherto unknown expressions for shape Hessians can be derived with little effort. The perspective of differential forms perfectly fits second-order boundary value problems (BVPs). We illustrate its power by deriving the shape derivatives of solutions to second-order elliptic BVPs with Dirichlet, Neumann and Robin boundary conditions. A new dual mixed variational approach is employed in the case of Dirichlet boundary conditions.