A hybrid variationally consistent homogenization approach based on Ritz's method

Multiscale approaches based on homogenization theory provide a suitable framework to incorporate information associated with a small‐scale (microscale) problem into the considered large‐scale (macroscopic) problem. In this connection, the present paper proposes a novel computationally efficient hybrid homogenization method. Its backbone is a variationally consistent FE² approach in which every aspect is governed by energy minimization. In particular, scale bridging is realized by the canonical principle of energy equivalence. As a direct implementation of the aforementioned variationally consistent FE² approach is numerically extensive, an efficient approximation based on Ritz's method is advocated. By doing so, the material parameters defining an effective macroscopic material model capturing the underlying microstructure can be efficiently computed. Furthermore, the variational scale bridging principle provides some guidance to choose a suitable family of macroscopic material models. Comparisons between the results predicted by the novel hybrid homogenization method and full field finite element simulations show that the novel method is indeed very promising for multiscale analyses.Copyright © 2013 John Wiley & Sons, Ltd.     

An energy momentum conserving algorithm using the variational formulation of visco‐plastic updates

In this paper we use the variational formulation of elasto‐plastic updates proposed by Ortiz and Stainier (Comput. Methods Appl. Mech. Eng. 1999; 171 :419– 444) in the context of consistent time integration schemes. We show that such a formulation is well suited to obtain a general expression of energy momentum conserving algorithms. Moreover, we present numerical examples that illustrate the efficiency of our developments. Copyright © 2005 John Wiley & Sons, Ltd.     

A variational formulation of constitutive models and updates in non‐linear finite viscoelasticity

The purpose of this article is to present a general framework for constitutive viscoelastic models in finite strain regime. The approach is qualified as variational since the constitutive updates obey a minimum principle within each load increment. The set of internal variables is strain‐based and employs, according to the specific model chosen, a multiplicative decomposition of strain into elastic and viscous components. The present approach shares the same technical procedures used for analogous models of plasticity or viscoplasticity, such as the solution of a minimization problem to identify inelastic updates and the use of exponential mapping for time integration. However, instead of using the classical decomposition of inelastic strains into amplitude and direction, we take advantage of a spectral decomposition that provides additional facilities to accommodate, into simple analytical expressions, a wide set of specific models. Moreover, appropriate choices of the constitutive potentials allow the reproduction of other formulations in the literature. The final part of the paper presents a set of numerical examples in order to explore the characteristics of the formulation as well as its applicability to usual large‐scale FEM analyses. Copyright © 2005 John Wiley & Sons, Ltd.     

Mixed variational principles and robust finite element implementations of gradient plasticity at small strains

This work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles. In contrast to classical local approaches to plasticity based on locally evolving internal variables, order parameter fields are taken into account governed by additional balance‐type PDEs including micro‐structural boundary conditions. This incorporates non‐local plastic effects based on length scales, which reflect properties of the material micro‐structure. We develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity, which is outlined for the simple model problem of 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 (FE) 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 FE implementations of gradient plasticity. This includes a rational method for the definition of elastic‐plastic‐boundaries in gradient plasticity along with a post‐processor that defines the plastic variables in the elastic range. We discuss alternative mixed FE designs of the coupled problem, including a local‐global solution strategy of short‐range and long‐range fields. This includes several new aspects, such as extended Q1P0‐type and Mini‐type finite elements for gradient plasticity. All methods are derived in a rigorous format from variational principles. Numerical benchmarks address advantages and disadvantages of alternative FE designs, and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity. Copyright © 2013 John Wiley & Sons, Ltd.     

Application of variational principles to the axial extension of a circular cylindical nonlinearly elastic membrane

In this paper stationary potential-energy and complementary-energy principles are formulated for boundary-value problems for compressible or incompressible nonlinearly elastic membranes, and full justification for adoption of the complementary principle is provided. The stationary principles are then extended to extremum principles, which provide upper and lower bounds on the energy functional associated with the solution of a given problem. The principles are then illustrated by their application to the nonlinear problem of the axially symmetric static deformation of an isotropic elastic membrane. In its undeformed natural configuration the membrane has the form of a circular cylindrical surface. The cylinder is subject to a prescribed (tensile) axial force with the ends of the cylinder constrained so that their radii remain constant. The alternative boundary condition in which the axial displacement of the ends is prescribed instead of the axial force is also considered. The extremum principles are applied first without restriction on the form of strain-energy function in order to obtain primitive bounds on the energy of Voigt and Reuss type commonly used in composite-material mechanics. Then, for particular forms of strain-energy function, specific bounds are obtained by selecting suitable trial deformation and stress fields and the bounds are optimized using a numerical procedure (which is readily adapted for other forms of strain-energy function). It is found that these bounds are very close and hence give a good estimate of the actual energy. The associated deformed geometry of the membrane is described together with the resulting principal stresses.     

Fluid structure interaction by means of variational multiscale reduced order models

A reduced order model designed by means of a variational multiscale stabilized formulation has been applied successfully to fluid-structure interaction problems in a strongly coupled partitioned solution scheme. Details of the formulation and the implementation both for the interaction problem and for the reduced models, for both the off-line and on-line phases, are shown. Results are obtained for cases in which both domains are reduced at the same time. Numerical results are presented for a semistationary and a fully transient case.     

On the formulation of closest‐point projection algorithms in elastoplasticity—part I: The variational structure

We present in this paper the characterization of the variational structure behind the discrete equations defining the closest‐point projection approximation in elastoplasticity. Rate‐independent and viscoplastic formulations are considered in the infinitesimal and the finite deformation range, the later in the context of isotropic finite‐strain multiplicative plasticity. Primal variational principles in terms of the stresses and stress‐like hardening variables are presented first, followed by the formulation of dual principles incorporating explicitly the plastic multiplier. Augmented Lagrangian extensions are also presented allowing a complete regularization of the problem in the constrained rate‐independent limit. The variational structure identified in this paper leads to the proper framework for the development of new improved numerical algorithms for the integration of the local constitutive equations of plasticity as it is undertaken in Part II of this work. Copyright © 2001 John Wiley & Sons, Ltd.     

动态流导法真空校准装置

动态流导法真空校准装置是真空计量的标准,可用于高真空和超高真空规的校准,真空度的校准范围为10-1～5×10-7 Pa,合成标准不确定度为0.69%～1.4%.     

Efficient modeling of localized material failure by means of a variationally consistent embedded strong discontinuity approach

This paper is concerned with a novel embedded strong discontinuity approach suitable for the analysis of material failure at finite strains. Focus is on localized plastic deformation particularly relevant for slip bands. In contrast to already existing models, the proposed implementation allows to consider several interacting discontinuities in each finite element. Based on a proper re‐formulation of the kinematics, an efficient parameterization of the deformation gradient is derived. It permits to compute the strains explicitly that improves the performance significantly. However, the most important novel contribution of the present paper is the advocated variational constitutive update. Within this framework, every aspect is naturally driven by energy minimization, i.e. all unknown variables are jointly computed by minimizing the stress power. The proposed update relies strongly on an extended principle of maximum dissipation. This framework provides enough flexibility for different failure types and for a broad class of non‐associative evolution equations. By discretizing the aforementioned continuous variational principle, an efficient numerical implementation is obtained. It shows, in addition to its physical and mathematical elegance, several practical advantages. For instance, the physical minimization principle itself specifies automatically and naturally the set of active strong discontinuities. Copyright © 2011 John Wiley & Sons, Ltd.     

A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns

The numerical modeling of dynamic failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in situations with complex crack topologies and demands the formulation of additional branching criteria. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. Following our recent works on quasi‐static modeling of phase‐field‐type brittle fracture, we propose in this paper a computational framework for diffusive fracture for dynamic problems that allows the simulation of complex evolving crack topologies. It is based on the introduction of a local history field that contains a maximum reference energy obtained in the deformation history, which may be considered as a measure of the maximum tensile strain in the history. This local variable drives the evolution of the crack phase field. Its introduction provides a very transparent representation of the balance equation that governs the diffusive crack topology. In particular, it allows for the construction of a very robust algorithmic treatment for elastodynamic problems of diffusive fracture. Here, we extend the recently proposed operator split scheme from quasi‐static to dynamic problems. In a typical time step, it successively updates the history field, the crack phase field, and finally the displacement field. We demonstrate the performance of the phase field formulation of fracture by means of representative numerical examples, which show the evolution of complex crack patterns under dynamic loading. Copyright © 2012 John Wiley & Sons, Ltd.     

A semi-inverse variational method for generating the bound state energy eigenvalues in a quantum system: the Dirac Coulomb type-equation

Exact eigenspectra and eigenfunctions of the Dirac quantum equation are established using the semi-inverse variational method. This method improves of a considerable manner the efficiency and accuracy of results compared with the other usual methods much argued in the literature. Some applications for different state configurations are proposed to concretize the method.     

Phase‐field modeling of ductile fracture at finite strains: A robust variational‐based numerical implementation of a gradient‐extended theory by micromorphic regularization

This work provides a robust variational‐based numerical implementation of a phase field model of ductile fracture in elastic–plastic solids undergoing large strains. This covers a computationally efficient micromorphic regularization of the coupled gradient plasticity‐damage formulation. The phase field approach regularizes sharp crack surfaces within a pure continuum setting by a specific gradient damage modeling with geometric features rooted in fracture mechanics. It has proven immensely successful with regard to the analysis of complex crack topologies without the need for fracture‐specific computational structures such as finite element design of crack discontinuities or intricate crack‐tracking algorithms. The proposed gradient‐extended plasticity‐damage formulation includes two independent length scales that regularize both the plastic response as well as the crack discontinuities. This ensures that the damage zones of ductile fracture are inside of plastic zones or vice versa and guarantees on the computational side a mesh objectivity in post‐critical ranges. The proposed setting is rooted in a canonical variational principle. The coupling of gradient plasticity to gradient damage is realized by a constitutive work density function that includes the stored elastic energy and the dissipated work due to plasticity and fracture. The latter represents a coupled resistance to plasticity and damage, depending on the gradient‐extended internal variables that enter plastic yield functions and fracture threshold functions. With this viewpoint on the generalized internal variables at hand, the thermodynamic formulation is outlined for gradient‐extended dissipative solids with generalized internal variables that are passive in nature. It is specified for a conceptual model of von Mises‐type elasto‐plasticity at finite strains coupled with fracture. The canonical theory proposed is shown to be governed by a rate‐type minimization principle, which fully determines the coupled multi‐field evolution problem. This is exploited on the numerical side by a fully symmetric monolithic finite element implementation. An important aspect of this work is the regularization towards a micromorphic gradient plasticity‐damage setting by taking into account additional internal variable fields linked to the original ones by penalty terms. This enhances the robustness of the finite element implementation, in particular, on the side of gradient plasticity. The performance of the formulation is demonstrated by means of some representative examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Solution of periodic group circular hole problems by using the series expansion variational method

This paper investigates the periodic group circular holes composed of infinite groups with numbering from j = ? ∞, …, ?2, ?1, 0, 1, 2, … to j = ∞ placed periodically in an infinite plate. The same loading condition and the same geometry are assumed for holes in all groups. The series expansion variational method (SEVM) is used for the solution of the periodic group circular hole problems. After using the SEVM, the boundary value problem is then reduced to an algebraic equation for the undetermined coefficients in the series expansion form, which is formulated on the central group. The influences on the central group from central group itself and many neighbouring groups are evaluated exactly. The influences on the central group from remote groups from j = ? ∞, ?(M + 2), ?(M + 1), M + 1, M + 2 to j = ∞ are approximately summed up into one term. This suggested technique is called the remainder estimation technique (RET) hereafter. It is proved from the computed results that the RET is very effective for the solution of the periodic group hole problems. Finally, several numerical examples are given and the interaction between the groups is addressed. Comparison between various sources of computation is presented. In the uniaxial tension in y‐direction, the stacking effect of the stacked groups is studied. Copyright © 2006 John Wiley & Sons, Ltd.     

A mathematical optimization methodology for the optimal design of a planar robotic manipulator

A general optimization methodology for the optimal design of robotic manipulators is presented and illustrated by its application to a realistic and practical three‐link revolute‐joint planar manipulator. The end‐effector carries out a prescribed vertical motion for which, respectively, the average torque requirement from electrical driving motors, and the electric input energy to the driving motors are minimized with respect to positional and dimensional design variables. In addition to simple physical bounds placed on the variables, the maximum deliverable torques of the driving motors and the allowable joint angles between successive links represent further constraints on the system. The optimization is carried out via a penalty function formulation of the constrained problem to which a proven robust unconstrained optimization method is applied. The problem of singularities (also known as degeneracy or lock‐up), which may occur for certain choices of design variables, is successfully dealt with by means of a specially proposed procedure in which a high artificial objective function value is computed for such ‘lock‐up trajectories’. Designs are obtained that are feasible and practical with reductions in the objective functions in comparison to that of arbitrarily chosen infeasible initial designs. Copyright © 1999 John Wiley & Sons, Ltd.     

A logarithmic‐exponential backward‐Euler‐based split of the flow rule for anisotropic inelastic behaviour at small elastic strain

A basic aspect of modern algorithmic formulations for large‐deformation hyperelastic‐based isotropic inelastic material models is the exponential backward‐Euler form of the algorithmic flow rule in the context of the multiplicative decomposition of the deformation gradient. Advantages of this approach in the isotropic context include the exact algorithmic fulfilment of inelastic incompressibility. The purpose of this short work is to show that such an algorithm can be formulated for anisotropic inelastic models as well under assumption of small elastic strain, i.e. for metals. In particular, the current approach works for both phenomenological anisotropy as well as for crystal plasticity. The major difference between the current and previous approaches lies in the fact that the elastic rotation is reduced algorithmically to a dependent internal variable, resulting in a smaller internal variable system. Copyright © 2006 John Wiley & Sons, Ltd.     

A quasi‐optimal coarse problem and an augmented Krylov solver for the variational theory of complex rays

The variational theory of complex rays (VTCR) is an indirect Trefftz method designed to study systems governed by Helmholtz‐like equations. It uses wave functions to represent the solution inside elements, which reduces the dispersion error compared with classical polynomial approaches, but the resulting system is prone to be ill‐conditioned. This paper gives a simple and original presentation of the VTCR using the discontinuous Galerkin framework, and it traces back the ill‐conditioning to the accumulation of eigenvalues near zero for the formulation written in terms of wave amplitude. The core of this paper presents an efficient solving strategy that overcomes this issue. The key element is the construction of a search subspace where the condition number is controlled at the cost of a limited decrease of attainable precision. An augmented LSQR solver is then proposed to solve efficiently and accurately the complete system. The approach is successfully applied to different examples. Copyright © 2015 John Wiley & Sons, Ltd.     

Evaluation of the T-stress and stress intensity factor for a cracked plate in general case using eigenfunction expansion variational method

This paper investigates the T-stress and stress intensity factor for a cracked plate in general case. In the general case, the shape of boundary and the applied loading are arbitrary. The eigenfunction expansion variational method (EEVM) is developed to evaluate the T-stress and stress intensity factor. For the traction boundary value problem, the EEVM is equivalent to the theorem of least potential energy in elasticity. Therefore, the EEVM possesses a clear physical meaning and it does not depend on any boundary collocation scheme. Several numerical examples are presented, which include: (1) a line crack in circular plate and (2) a line crack in rectangular plate. Numerical examination for convergence in an example is carried out.     

The special‐linear update: An application of differential manifold theory to the update of isochoric plasticity flow rules

The evolution of plastic deformations in metals, governed by incompressible flow rules, has been traditionally solved using the exponential mapping. However, the accurate calculation of the exponential mapping and its tangents may result in computationally demanding schemes in some cases, while common low‐order approximations may lead to poor behavior of the constitutive update because of violation of the incompressibility condition. Here, we introduce the special‐linear (SL) update for isochoric plasticity, a flow‐rule integration scheme based on differential manifolds concepts. The proposed update exactly enforces the plastic incompressibility condition while being first‐order accurate and consistent with the flow rule, thus bearing all the desirable properties of the now standard exponential mapping update. In contrast to the exponential‐mapping update, we demonstrate that the SL update can drastically reduce the computing time, reaching one order of magnitude speed‐ups in the calculation of the update tangents. We demonstrate the applicability of the update by way of simulation of single‐crystal plasticity uniaxial loading tests. We anticipate that the SL update will open the way to efficient constitutive updates for the solution of complex multiscale material models, thus making it a very promising tool for large‐scale simulations. Copyright © 2013 John Wiley & Sons, Ltd.     

聚四氟乙烯包覆铝粉烧结的模拟与分析

为了分析烧结过程对聚四氟乙烯(PTFE)包覆Al粉性能的影响,利用分子动力学手段计算了298 K和678 K时PTFE在Al2O3(001)、(010)及(100)晶面6×6晶层的结合能,利用耗散粒子动力学手段模拟了298 K和678 K时Al2O3/PTFE在不同时刻的介观状态。计算及模拟结果表明:298 K时,PTFE在Al2O3(001)、(010)及(100)晶面的6×6晶层的结合能分别为2 782.67、5 582.97及4 634.32 k J/mol;678 K时PTFE与Al2O3(001)、(010)及(100)晶面的结合能分别是2 835.29、5 537.54及4 608.49 k J/mol。低温时,PTFE和Al2O3混溶性差,两种物质发生明显分相;高温时,PTFE和Al2O3混溶性较好,没有发生明显的分相。烧结过程有助于PTFE在Al2O3中的扩散,同时还可以提高聚合物与Al2O3的混溶性,但对PTFE包覆Al粉的强度影响不大。     

Optical properties of alkali-earth atoms and Na2 calculated by GW and Bethe–Salpeter equations

Starting from the GW approximation (GWA) beyond density functional theory and solving the eigenvalue problem associated with the Bethe–Salpeter equation which accounts for the excitonic effect, I have determined optical properties of isolated Be, Mg and Ca atoms and Na₂. In the representation of single electron wave functions, I have used the all-electron mixed basis approach in which both plane waves and atomic orbitals are used as a basis set. The resulting quasiparticle energies and optical absorption spectra are compared with available experimental data.