Multigrid fictitious boundary and grid deformation methods for flow-induced rotation of wing

Numerical simulations of flow-induced rotation of wing by multigrid fictitious boundary and grid deformation methods are presented. The flow is computed by a special ALE formulation with a multigrid finite element solver. The solid wing is allowed to move freely through the computational mesh which is adaptively aligned by a special mesh deformation method. The advantage of this approach is that no expensive remeshing has to be performed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Simulation of brittle crack growth in functionally graded materials

We consider a thermodynamic consistent framework for crack propagation by applying a dissipation inequality to a time dependent migrating control volume. The direction of crack growth is obtained in terms of material forces as a result of the principle of maximum dissipation. In the numerical implementation a staggered algorithm – deformation update for fixed geometry followed by geometry update for fixed deformation – is employed within each time increment. The corresponding mesh is generated by combining Delaunay triangulation with local mesh refinement. A numerical example with inhomogeneous material properties illustrates the capability of the resulting algorithm. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

SPH method applied to compression of solid materials for a variety of loading conditions

Smoothed Particle Hydrodynamics (SPH) is a numerical method that does not use a mesh or grid when solving a set of partial differential equations. This makes it particularly useful in application to solid mechanics problems where the sample undergoes large deformation. Whereas mesh-based methods have difficulty when the sample becomes severely distorted, SPH naturally deals with this important engineering scenario. We implement the SPH method for compressional deformation of solid samples and focus on uniaxial, biaxial and triaxial loading. We develop numerical procedures that naturally deal with these three different sets of boundary conditions and apply it to both small and larger strains in elastic and more complex materials. The method is shown to be robust up to large strains of 30%. Under uniaxial loading, a cylindrical sample tends to deform by bulging while under triaxial loading the cylindrical sample will remain cylindrical, but the diameter of the sample increases accordingly.     

Optimal orientation of anisotropic solids

Results are presented for finding the optimal orientation ofan anisotropic elastic material. The problem is formulated asminimizing the strain energy subject to rotation of the materialaxes, under a state of uniform stress. It is shown that a stationaryvalue of the strain energy requires the stress and strain tensorsto have a common set of principal axes. The new derivation ofthis well-known coaxiality condition uses the six-dimensionalexpression of the rotation tensor for the elastic moduli. Usingthis representation it is shown that the stationary conditionis a minimum or a maximum if an explicit set of conditions issatisfied. Specific results are given for materials of cubic,transversely isotropic (TI) and tetragonal symmetries. In eachcase the existence of a minimum or maximum depends on the signof a single elastic constant. The stationary (minimum or maximum)value of energy can always be achieved for cubic materials.Typically, the optimal orientation of a solid with cubic materialsymmetry is not aligned with the symmetry directions. Expressionsare given for the optimal orientation of TI and tetragonal materials,and are in agreement with results of Rovati and Taliercio obtainedby a different procedure. A new concept is introduced: the straindeviation angle, which defines the degree to which a state ofstress or strain is not optimal. The strain deviation angleis zero for coaxial stress and strain. An approximate formulais given for the strain deviation angle which is valid for materialsthat are weakly anisotropic.     

Nature of the friction deformation of the surface layers of polymethyl methacrylate

The surface deformation of amorphous thermoplastics (polymethyl methacrylate) by a spherical steel indentor has been investigated at various sliding velocities. Small velocities correspond to elastic and forced-elastic deformation of the surface layers and asperities. At temperatures corresponding to the high-elastic state the deformed surface layer completely recovers its shape. As the sliding velocity increases, the forced-elastic deformation is localized in a thinner layer of plastic. Starting from a certain velocity, depending on the temperature and the activation energy for transition of the chain segments from one equilibrium position to another in the process of thermal motion, the deformation of the surface layers and asperities becomes purely elastic. In the event of elastic deformation at pressures above a certain value the surface layer of plastic suffers brittle fracture in the tensile zone behind the indentor.Mekhanika Polimerov, Vol. 4, No. 1, pp. 90–94, 1968.     

Simulation of Atomic Force Microscopy for investigating BaTiO3 and LiMn2O4 nanostructures based on Phase Field Approach

Atomic Force Microscopy (AFM) probes the surface features of specimens using an extremely sharp tip scanning the sample surface while the force is applied. AFM is also widely used for investigating the electrically non-conductive materials by applying an electric potential on the tip. Piezoresponse Force Microscopy (PFM) and Electrochemical Strain Microscopy (ESM) are variants of AFM for different materials. Both PFM and ESM signals are obtained by observing the displacement of the tip when applying electric fields during the scanning process. The PFM technique is based on converse piezoelectric effect of ferroelectrics and the ESM technique is based on electrochemical coupling in solid ionic conductors. In this work, two continuum-mechanical formulations for simulation of PFM and ESM are discussed. In the first model, for PFM simulation, a phase field approach based on the Allen-Cahn equation for non-conserved order parameters is employed for ferroelectrics. Here, the polarization vector is chosen as order parameter. Since ferroelectrics have highly anisotropic properties, this model accounts for transversely isotropic symmetry using an invariant formulation. The polarization switching behavior under the electric field will be discussed with some numerical examples. In the simulation of ESM, we employ a constitutive model based on the work of Bohn et al. [8] for the modeling of lithium manganese dioxide LiMn₂O₄ (LMO). It simulates the deformation of the LMO particle according to an applied voltage and the evolution of lithium concentration after removing a DC pulse. The modeling results are compared to experimental data. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Latitudinally deforming rotating sphere

In this study, we consider a sphere with a surface that is fully covered by a stretchable elastic material. The radius of the sphere is fixed and it is also rotating about its radial axis. We investigate how the axisymmetric motion of a triggered fluid flow around the sphere is affected by the presence of both sphere rotation and latitudinal stretching. Considering that the deformation over the sphere commences at the pole, the problem is formulated such that the fluid flow near the pole is similar to the induced flow due to a linearly stretchable rotating disk, which has been described well in previous studies. When the rotation is omitted, the flow develops two-dimensionally under the action of pure stretching; otherwise, a three-dimensional axisymmetric fluid flow occurs, which is computed at each latitudinal angle both numerically and using a perturbation approach. The solution with wall deformation is different from the traditional character of the solution due to a solely rotating sphere. This solution is then used to compute the surface shears due to the physical drag and torque acting over the sphere. The contribution of wall stretching reduces the drag, whereas high rotation suppresses the effects of stretching to enhance the drag. More torque is required to rotate the sphere when both stretching and rotation mechanisms are in action.     

ALTERING CONNECTIVITY WITH LARGE DEFORMATION MESH FOR LAGRANGIAN METHOD AND ITS APPLICATION IN MULTIPLE MATERIAL SIMULATION

A new approach for treating the mesh with Lagrangian scheme of finite volume method is presented. It has been proved that classical Lagrangian method is difficult to cope with large deformation in tracking material particles due to severe distortion of cells, and the changing connectivity of the mesh seems especially attractive for solving such issues. The mesh with large deformation based on computational geometry is optimized by using new method. This paper develops a processing system for arbitrary polygonal unstructured grid, the intelligent variable grid neighborhood technologies is utilized to improve the quality of mesh in calculation process, and arbitrary polygonal mesh is used in the Lagrangian finite volume scheme. The performance of the new method is demonstrated through series of numerical examples, and the simulation capability is efficiently presented in coping with the systems with large deformations.     

A constrained optimization approach to finite element mesh smoothing

The quality of a finite element solution has been shown to be affected by the quality of the underlying mesh. A poor mesh may lead to unstable and/or inaccurate finite element approximations. Mesh quality is often characterized by the “smoothness” or “shape” of the elements (triangles in 2-D or tetrahedra in 3-D). Most automatic mesh generators produce an initial mesh where the aspect ratio of the elements are unacceptably high. In this paper, a new approach to produce acceptable quality meshes from a topologically valid initial mesh is presented. Given an initial mesh (nodal coordinates and element connectivity), a “smooth” final mesh is obtained by solving a constrained optimization problem. The variables for the iterative optimization procedure are the nodal coordinates (excluding, the boundary nodes) of the finite element mesh, and appropriate bounds are imposed on these to prevent an unacceptable finite element mesh. Examples are given of the application of the above method for 2- and 3-D meshes generated using automatic mesh generators. Results indicate that the new method not only yields better quality elements when compared with the traditional Laplacian smoothing, but also guarantees a valid mesh unlike the Laplacian method.     

Modelling of metal forging using SPH

In solid metal forming processes, such as forging, large distortions in the material present challenging problems for numerical simulation using grid based methods. Computations invariably fail after some level of mesh distortion is reached unless suitable re-meshing is implemented to cope with the mesh distortion arising from the material deformation. The issue of mesh distortion and the subsequent re-meshing are topics of much research for grid based methods. These problems can be overcome by using a mesh-less numerical framework. In this paper, the application of a mesh-less method called Smoothed Particle Hydrodynamics (SPH) for modelling three-dimensional complex forging processes is demonstrated. It is shown that SPH is a useful simulation method for obtaining insights into the material deformation and flow pattern during forging of realistic industrial components. The effect of process parameters and material properties on the quality of the forged component is evaluated via SPH simulations. This includes the determination of forging force required for adequate die filling which is an important criterion for die designs. Material hardening, controlled by the degree of heat treatment, is found to have a profound effect on the material deformation pattern and the final product. Forging defects such as incomplete die filling, asymmetry in forged components, flashing and lap formation are shown to be predicted by SPH. SPH can thus potentially be used both for assessment of the quality of forged products and evaluation of prototype forging system designs.     

液气界面张力垂直分量引起的基底弹性变形

Young方程是毛细理论和润湿的重要方程之一．但是,该方程只描述了3个界面张力的水平分量之间的平衡与接触角的关系,而对液气界面张力垂直分量未作任何描述．现在,随着软材料的广泛应用,该垂直分量将引起基底的表面变形,并在微流体系统的制造过程中起到重要作用,这已是该研究领域的共识．综述了关于表面变形这一问题在理论分析,实验研究和数值模拟等方面取得的进展．而且,还讨论了由垂直分量引起的表面变形对液滴润湿和铺展行为、微悬臂梁的弯曲、弹性毛细现象、电弹性毛细现象等的影响．不仅对该问题的历史发展和目前的研究进展进行了简单的综述,并且也针对后续的研究提出了几点建议．     

Boundary conforming Delaunay mesh generation

A boundary conforming Delaunay mesh is a partitioning of a polyhedral domain into Delaunay simplices such that all boundary simplices satisfy the generalized Gabriel property. It’s dual is a Voronoi partition of the same domain which is preferable for Voronoi-box based finite volume schemes. For arbitrary 2D polygonal regions, such meshes can be generated in optimal time and size. For arbitrary 3D polyhedral domains, however, this problem remains a challenge. The main contribution of this paper is to show that boundary conforming Delaunay meshes for 3D polyhedral domains can be generated efficiently when the smallest input angle of the domain is bounded by arccos 1/3 ≈ 70.53°. In addition, well-shaped tetrahedra and an appropriate mesh size can be obtained. Our new results are achieved by reanalyzing a classical Delaunay refinement algorithm. Note that our theoretical guarantee on the input angle (70.53°) is still too strong for many practical situations. We further discuss variants of the algorithm to relax the input angle restriction and to improve the mesh quality.     

Assessing Logo programming among Jordanian seventh grade students through turtle geometry

The present study is concerned with assessing Logo programming experiences among seventh grade students. A formal multiple-choice test and five performance tasks were used to collect data. The results provided that students’ performance was better than the expected score by the probabilistic laws, and a very low correlation between their Logo programming performance and school mathematics achievement was revealed. Most of the made misconceptions were due to geometrical aspects rather than Logo primitives, and were concentrated on the angle of rotation, the angle of complete rotation and the angle of regular polygon. In addition, students’ problem-solving ability was limited while conducting some Logo programming tasks, and acceptable in others. In regard to the results, it is recommended that teaching Logo programming should be used in different contexts that enhance students’ learning, and develop problem-solving processes.     

Elastoplastic torsion of a cylindrical rod for finite deformations

A solution of the problem of the torsion of a cylindrical rod was obtained in /1/ for a general, isotropic, incompressible elastic material. The present paper gives an analytical solution of the elastoplastic torsion problem for finite deformations, written in terms of quadratures of elliptic functions. The non-linear kinematics of elastoplastic deformation is introduced into the defining equations with the help of a multiplicative decomposition of the deformation gradient into elastic and plastic components /2, 3/. The elastic deformation and rate of plastic deformation are related to the state of stress of the body, in accordance with the defining Mooney-Rivlin equations /4/ and the law of flow for finite deformations associated with the Tresca yield condition /5/. A non-linear first-order partial differential equation and the initial data at the elastoplastic boundary are obtained in order to determine the angle of rotation within the plastic zone of the basis formed from the eigenvectors of the stress tensor, relative to the radial direction. The integration of the resulting equation is reduced to determining the general integral of the Ricatti equation with right-hand side determined from the angular velocity of flow of the material within the plastic zone. It is shown that neglecting the finiteness of the deformation leads to too high an estimate of the rigidity of the rod.     

The dual nature of the transverse vibrations of an elastic rod

The transverse vibrations of an elastic rod, to one of which displacements are applied while the other end is free, are investigated. It is assumed that the propagation velocity of the perturbations in the rod is finite. The unperturbed part performed rotational motion around the centre line. The angle of rotation is expressed by the angle of curvature of the centre line of the perturbed part of the rod. Two types of elastic vibrations are obtained: (1) the rod vibrates elastically due to displacements applied at the end, and (2) when performing rotational motion elastic vibrations and additional forces occur in the rod due to elasticity [1].     

Deformation of sandwich panels in cylindrical bending by point forces. 1. Analytical construction

A refined model for bending of a three-layered panel with a soft filler is proposed. The modified model permits us to consider the asymmetry of elastic properties and thickness of the outer layer relative to the middle plane of the panel in a composite sandwich structure. In constructing the deformation mechanism, a heterogeneous kinematic model was adopted, which, in contrast to the assumptions for the deformation of the whole stack of layers, features four degrees of displacement freedom permitting consideration of the separate nature of the deformation of the outer layers in bending and of the intermediate layer in transverse compression combined with shear. This approach is postulated according to an energetic evaluation of the deformation of the layers [2]. The specific features of the stress from point forces in cylindrical bending are considered using the operational Laplace method, which avoids the additional difficulties in analyzing the solution convergence arising when it is represented by a series of eigenfunctions of the boundary value problem. The fundamental functions of a twelfth-order set of equations are used to construct the boundary problem reduced to a Cauchy problem. Various boundary effects of the point stress are described using a generalized Dirac function. Variants are examined for the limiting transformation of the model parameters leading to a qualitative change in its kinematics and the corresponding simplified bending models. Institute of Polymer Mechanics, Latvian Academy of Sciences, LV-1006 Riga, Latvia. Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 588–611, September–October, 1996.     

Justification of the Kirchhoff hypotheses and error estimation for two-dimensional models of anisotropic and inhomogeneous plates, including laminated plates

Asymptotic analysis of the problem describing deformation ofa thin cylindrical plate with clamped lateral side is performed.The problem is considered under the most general statement withthe plate being laminated and consisting of an arbitrary numberof nonhomogeneous and anisotropic (21 elastic moduli) layers.Explicit integral representations of the differential operatorswhich form the two-dimensional model of the plate are derived.In the case when the elastic moduli of each of the layers areconstant, these integral representations turn into algebraicones. The asymptotic procedure is justified with the help ofa weighted inequality of Korn's type. The error estimates obtainedgive a rigorous mathematical proof of both of Kirchhoff's hypotheses(kinematic and static) and shed light on the well-known intrinsicinconsistency of two of the hypotheses.     

An efficient quantitative ultrasonic elastography based on variational h-refinement

Quantitative elastography is a method to visualise a stiffness distribution. It is motivated by the observation that changes in mechanical properties of soft tissue mostly include important diagnostic information. With an ultrasound–elastography system, the displacement field can be calculated from the pre– and post–deformation image. Using the assumption that the material is elastic, isotropic and nearly incompressible, the distribution of shear modulus is determined by solving an inverse problem. While common approaches use a constant mesh, a variational h–refinement is in the focus of this work. In doing so, the efficiency and accuracy of the determination can be increased. The presented results are generated by using numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)