Quadratic Shape Research Articles

Geometrically nonlinear polygonal finite element analysis of functionally graded porous plates

In this study, an efficient polygonal finite element method (PFEM) in combination with quadratic serendipity shape functions is proposed to study nonlinear static and dynamic responses of functionally graded (FG) plates with porosities. Two different porosity types including even and uneven distributions through the plate thickness are considered. The quadratic serendipity shape functions over arbitrary polygonal elements including triangular and quadrilateral ones, which are constructed based on a pairwise product of linear shape functions, are employed to interpolate the bending strains. Meanwhile, the shear strains are defined according to the Wachspress coordinates. By using the Timoshenko's beam to interpolate the assumption of the strain field along the edges of polygonal element, the shear locking phenomenon can be naturally eliminated. Furthermore, the C0–type higher-order shear deformation theory (C0–HSDT), in which two additional variables are included in the displacement field, significantly improves the accuracy of numerical results. The nonlinear equations of static and dynamic problems are solved by Newton–Raphson iterative procedure and by Newmark's integration scheme in association with the Picard methods, respectively. Through various numerical examples in which complex geometries and different boundary conditions are involved, the proposed approach yields more stable and accurate results than those generated using other existing approaches.

An adaptive higher order finite element model and modal energy for the vibration of a traveling string

A nonlinear equation describing the transverse vibration of an axially traveling string with constant and time-varying length is obtained by developing a new finite element model described by quadratic shape functions. A novel nonlinear coordinate transform is introduced with regard to its nonlinear terms. Subsequently, a new hybrid Newmark-beta/time varying degree of freedom method, which can adjust the element number automatically according to the change of string length, is proposed to improve accuracy. The proposed method as well as normal numerical methods are compared with an analytical solution. Results show that the proposed method is in good agreement with the Newmark-beta method for the case of small variations in string length, whilst it is superior in accuracy to the latter in the case of large length variations. Complex mode theory is adopted firstly to obtain the modal components as well as subsequently the modal energy for a traveling string. A phenomenon is observed where the free vibration energy leaks from one mode to the others in a traveling string. The higher the speed of translation and the modal order, the more energy that is leaked into the modes close to the initially excited mode.

Subdivision surfaces with isogeometric analysis adapted refinement weights

Subdivision surfaces provide an elegant isogeometric analysis framework for geometric design and analysis of partial differential equations defined on surfaces. They are already a standard in high-end computer animation and graphics and are becoming available in a number of geometric modelling systems for engineering design. The subdivision refinement rules are usually adapted from knot insertion rules for splines. The quadrilateral Catmull–Clark scheme considered in this work is equivalent to cubic B-splines away from extraordinary, or irregular, vertices with other than four adjacent elements. Around extraordinary vertices the surface consists of a nested sequence of smooth spline patches which join C1 continuously at the point itself. As known from geometric design literature, the subdivision weights can be optimised so that the surface quality is improved by minimising short-wavelength surface oscillations around extraordinary vertices. We use the related techniques to determine weights that minimise finite element discretisation errors as measured in the thin-shell energy norm. The optimisation problem is formulated over a characteristic domain and the errors in approximating cup- and saddle-like quadratic shapes obtained from eigenanalysis of the subdivision matrix are minimised. In finite element analysis the optimised subdivision weights for either cup- or saddle-like shapes are chosen depending on the shape of the solution field around an extraordinary vertex. As our computations confirm, the optimised subdivision weights yield a reduction of 50% and more in discretisation errors in the energy and L2 norms. Although, as to be expected, the convergence rates are the same as for the classical Catmull–Clark weights, the convergence constants are improved.

Three-dimensional fracture instability of a displacement-weakening planar interface under locally peaked nonuniform loading

We consider stability of fracture on a three-dimensional planar interface subjected to a loading stress that is locally peaked spatially, the level of which increases quasi-statically in time. Similar to the earlier study on the two-dimensional case (Uenishi and Rice, 2003; Rice and Uenishi, 2010), as the loading stress increases, a crack, or a region of displacement discontinuity (opening gap in tension or slip for shear fracture), develops on the interface where the stress is presumed to decrease according to a displacement-weakening constitutive relation. Upon reaching the instability point at which no further quasi-static solution for the extension of the crack on the interface exists, dynamic fracture follows. For the investigation of this instability point, we employ a dimensional analysis as well as an energy approach that gives a Rayleigh–Ritz approximation for the dependence of crack size and maximum displacement discontinuity on the level and quadratic shape of the loading stress distribution. We show that, if the linear displacement-weakening law is applied and the crack may be assumed of an elliptical form, the critical crack size at instability is independent of the curvature of the loading stress distribution and it is of the same order for all two- and three-dimensional cases.

Numerical Solution of a Wave Propagation Problem Along Plate Structures Based on the Isogeometric Approach

Ultrasonic guided waves propagating along large structures have great potential as a nondestructive evaluation method. In this context, it is very important to obtain the dispersion curves, which depend on the cross-section of the structure. In this paper, we compute dispersion curves along infinite isotropic plate-like structures using the semi-analytical method (SAFEM) with an isogeometric approach based on B-spline functions. The SAFEM method leads to a family of generalized eigenvalue problems depending on the wave number. For a prescribed wave number, the solution of this problem consists of the nodal displacement vector and the frequency of the guided wave. In this work, the results obtained with B-splines shape functions are compared to the numerical SAFEM solution with quadratic Lagrange shape functions. Advantages of the isogeometric approach are highlighted and include the smoothness of the displacement field components and the computational cost of solving the corresponding generalized eigenvalue problems. Finally, we investigate the convergence of Lagrange and B-spline approaches when the number of degrees of freedom grows. The study shows that cubic B-spline functions provide the best solution with the smallest relative errors for a given number of degrees of freedom.

Symmetry Breaking in Photonic Crystals: On-Demand Dispersion from Flatband to Dirac Cones.

We demonstrate that symmetry breaking opens a new degree of freedom to tailor energy-momentum dispersion in photonic crystals. Using a general theoretical framework in two illustrative practical structures, we show that breaking symmetry enables an on-demand tuning of the local density of states of the same photonic band from zero (Dirac cone dispersion) to infinity (flatband dispersion), as well as any constant density over an adjustable spectral range. As a proof of concept, we demonstrate experimentally the transformation of the very same photonic band from a conventional quadratic shape to a Dirac dispersion, a flatband dispersion, and a multivalley one. This transition is achieved by finely tuning the vertical symmetry breaking of the photonic structures. Our results provide an unprecedented degree of freedom for optical dispersion engineering in planar integrated photonic devices.

Ontogeny, But Not Sexual Dimorphism, Drives the Intraspecific Variation of Quadrate Morphology in Hemidactylus turcicus (Squamata: Gekkonidae)

Abstract The functional components of the lizard skull are divided into the chondrocranial braincase, protective dermatocranium, lower jaw, and hyobranchial apparatus. These regions are interconnected and become operational through the quadrate, a bone critical for cranial biomechanics and support of the peripheral auditory system. The quadrate is a complex and variable structure in squamates; however, neither the intraspecific nor interspecific variation of this element has been studied in detail. We investigated the intraspecific variation of quadrate morphology within Hemidactylus turcicus with the use of cleared and double-stained specimens and high-resolution x-ray microcomputed tomography. Our objectives were to quantify quadrate shape and the degree of intraspecific variation within this element with the use of 2-D and 3-D geometric morphometric analyses and investigate if this variation is driven by ontogeny and/or sexual dimorphism. Our results demonstrate that ontogeny, but not sexual shape dimo...

Thermo‐mechanical modeling of turbine blades taking into account structural defects

AbstractIn turbine blades of aero‐engines typical defects are cracks due to high mechanical and thermal loads. The extended finite element method (XFEM) is used for simulations of fracture mechanics problems with cracks. Discontinuities in the displacement and temperature field are allowed and the crack opening displacement and crack tip stress field are reproduced accurately. Since crack closure and non‐physical penetration of the crack surfaces may occur under certain load conditions, it becomes necessary to enforce the non‐penetration condition for crack surfaces. This contact formulation is assumed to be frictionless. The node‐to‐segment approach proposed in [3] is extended to ten‐node tetrahedral elements with quadratic shape functions. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Taxonomic decomposition of the latitudinal gradient in species diversity of North American floras

AbstractAimTo test the latitudinal gradient in plant species diversity for self‐similarity across taxonomic scales and amongst taxa.LocationNorth America.MethodsWe used species richness data from 245 local vascular plant floras to quantify the slope and shape of the latitudinal gradients in species diversity (LGSD) across all plant species as well as within each family and order. We calculated the contribution of each family and order to the empirical LGSD.ResultsWe observed the canonical LGSD when all plants were considered with floras at the lowest latitudes having, on average, 451 more species than floras at the highest latitudes. When considering slope alone, most orders and families showed the expected negative slope, but 31.7% of families and 27.7% of orders showed either no significant relationship between latitude and diversity or a reverse LGSD. Latitudinal patterns of family diversity account for at least 14% of this LGSD. Most orders and families did not show the negative slope and concave‐down quadratic shape expected by the pattern for all plant species. A majority of families did not make a significant contribution in species to the LGSD with 53% of plant families contributing little to nothing to the overall gradient. Ten families accounted for more than 70% of the gradient. Two families, the Asteraceae and Fabaceae, contributed a third of the LGSD.Main ConclusionsThe empirical LGSD we describe here is a consequence of a gradient in the number of families and diversification within relative few plant families. Macroecological studies typically aim to generate models that are general across taxa with the implicit assumption that the models are general within taxa. Our results strongly suggest that models of the latitudinal gradient in plant species richness that rely on environmental covariates (e.g. temperature, energy) are likely not general across plant taxa.

English

English

Static phantom wormholes of finite size

In this paper we derive new static phantom traversable wormholes by assuming a shape function with a quadratic dependence on the radial coordinate $r$. We mainly focus our study on wormholes sustained by exotic matter with positive energy density (as seen by any static observer) and a variable equation of state ${p}_{r}/\ensuremath{\rho}<\ensuremath{-}1$, dubbed ``phantom matter.'' Among phantom wormhole spacetimes extending to infinity, we show that a quadratic shape function allows us to construct static spacetimes of finite size, composed of a phantom wormhole connected to an anisotropic spherically symmetric distribution of dark energy. The wormhole part of the full spacetime does not fulfill the dominant energy condition, while the dark energy part does.

Quadratic serendipity finite elements over convex polyhedra

SummaryThe conventional approach to construct quadratic elements for an n‐sided polygon will yield n(n+1)/2 shape functions, which increases the computational effort. It is well known that the serendipity elements based on isoparametric formulation suffers from mesh distortion. Floater and Lai proposed a systematic way to construct higher‐order approximations over arbitrary polygons using the generalized barycentric and triangular coordinates. This approach ensures 2n shape functions with nodes only on the boundary of the polygon. In this paper, we extend the polygonal splines approach to 3 dimensions and construct serendipity shape functions over hexahedra and convex polyhedra. This is done by expressing the shape functions using the barycentric coordinates and the local tetrahedral coordinates. The quadratic shape functions possess Kronecker delta property and satisfy constant, linear, and quadratic precision. The accuracy and the convergence properties of the quadratic serendipity shape elements are demonstrated with a series of standard patch tests. The numerical results show that the quadratic serendipity elements pass the patch test, yield optimal convergence rate, and can tolerate extreme mesh distortion.

Numerical effects on notch fatigue strength assessment of non-welded and welded components

Local fatigue strength assessment based on threshold values obtained by linear-elastic notch stress calculation is commonly utilized due to its applicability to complex geometries with an acceptable effort. As the result of a finite element computation is affected by the element type and mesh refinement, this paper investigates the numerical influence on the notch stress based fatigue strength assessment of non-welded and welded components. Based on extensive finite element studies employing the software package Abaqus it is concluded that quadratic shape functions with a number of sixteen for non-welded parts and twelve elements over a semicircle for welded joints should be at least applied to minimize the numerical impact.

A family of exponentially-gradient elements for numerical computation of singular boundary value problems

In this paper, a new family of single-parameter exponentially gradient elements (EG-elements) are introduced, which can be used in various numerical procedures such as boundary and finite element methods. These elements have the ability to accurately interpolate the unknown values in regions, where either high gradient or singularity of the unknown field occurs. The shape functions of two-dimensional EG-elements are high gradient at either a corner of the element or at an edge of the element. Another advantage of this element is that the regular quadratic shape functions are obtained as a special case of EG-elements by adjusting the single parameter of the element, which allows this element to be used as regular Lagrange quadratic element, where it is appropriate. Some mixed boundary value problems are solved with the use of EG-elements in a boundary element program to show the capability of these elements for capturing the solution with less number of elements and higher accuracy.

Quadratic relationships for a strip small-period array

A strip array with a period small with respect to the wavelength is considered. The far field of this array is described by the nonlocal equivalent boundary conditions. It is shown with the help of the active power theorem that a near field transferring the active power along metal strips exists in the strip. The relationships connecting the far field on the strip surface with the active power of the near field are derived. It is demonstrated with the help of the Lorentz lemma that the near field has a nonzero quadratic shape and the relationships connecting it with the far field are obtained. A method of determination of the far field components on the array surface that are normal to the array plane is proposed. This method allows one to take into account the finite values of the quadratic forms.

Analysis of pressure variation of fluid in bounded circular reservoirs under the constant pressure outer boundary condition

In this work, we have investigated the well pressure distribution in a bounded circular reservoir under the condition of constant pressure outer boundaries. The diffusivity equation was used in the analysis. The finite element technique, using Lagrange quadratic shape elements was employed to carry out the analysis over the cross-section of the reservoir which involves discretizing the domain into finite element, analysing these finite element, assembling the results from the analysis of the analysed finite element, imposing the boundary conditions and finally, getting the results that represent the entire domain. The results obtained where shown in a log log plot (dimensionless pressure against dimensionless time) for dimensionless radii of 1 to 1,000,000 in log cycles. It was shown that the relationship between dimensionless pressure and dimensionless time was linear whose slope was zero. The result obtained at the wellbore was compared with the results obtained by Van Everdigen and Hurst. It was shown that there was a strong positive correlation between the results. The result obtained from the analysis also shows the pressure variation outside wellbore of the same reservoir. It is important to note that solutions from existing literature only state the pressure at the wellbore at a particular time but this work predicts the pressure variation in the entire reservoir from the wellbore to the external boundary at the same time. http://dx.doi.org/10.4314/njt.v36i1.20

Study on the basic characteristics of a noncontact air conveyor for large glass sheets

Recently, the development of displays requires large, thin glass substrates, and air film conveyors equipped with porous pads have been developed to transport the substrates in a noncontact state. In this study, a contactless air film system with aligned porous pads is proposed. Experiments and analysis of its basic characteristics are conducted. First, it is confirmed that the flow rate characteristics of the porous pads can be represented by a modified Forchheimer equation with the determined permeability and inertial coefficient. Then, a theoretical model based on the Reynolds equation is established to analyze the film pressure distribution. The results indicate a parabola-like shape distributed along the radial direction in the porous area and an approximately linear curve in the periphery area. Furthermore, the experimental results indicate that the bearing force and static stiffness increase if the clearance decreased with a given flow rate. Finally, a numerical calculation performed by solving the coupling equation of elasticity and the modified Reynolds equation shows an approximate quadratic shape in the deformation, and the experimental results supported this to a certain extent.

Numerical Solution of a Wave Propagation Problem Along Plate Structures Based on the Isogeometric Approach

Ultrasonic guided waves propagating along large structures have great potential as a nondestructive evaluation method. In this context, it is very important to obtain the dispersion curves, which depend on the cross-section of the structure. In this paper, we compute dispersion curves along infinite isotropic plate-like structures using the semi-analytical method (SAFEM) with an isogeometric approach based on B-spline functions. The SAFEM method leads to a family of generalized eigenvalue problems depending on the wave number. For a prescribed wave number, the solution of this problem consists of the nodal displacement vector and the frequency of the guided wave. In this work, the results obtained with B-splines shape functions are compared to the numerical SAFEM solution with quadratic Lagrange shape functions. Advantages of the isogeometric approach are highlighted and include the smoothness of the displacement field components and the computational cost of solving the corresponding generalized eigenval...

SVBRDF-Invariant Shape and Reflectance Estimation from a Light-Field Camera.

Light-field cameras have recently emerged as a powerful tool for one-shot passive 3D shape capture. However, obtaining the shape of glossy objects like metals or plastics remains challenging, since standard Lambertian cues like photo-consistency cannot be easily applied. In this paper, we derive a spatially-varying (SV)BRDF-invariant theory for recovering 3D shape and reflectance from light-field cameras. Our key theoretical insight is a novel analysis of diffuse plus single-lobe SVBRDFs under a light-field setup. We show that, although direct shape recovery is not possible, an equation relating depths and normals can still be derived. Using this equation, we then propose using a polynomial (quadratic) shape prior to resolve the shape ambiguity. Once shape is estimated, we also recover the reflectance. We present extensive synthetic data on the entire MERL BRDF dataset, as well as a number of real examples to validate the theory, where we simultaneously recover shape and BRDFs from a single image taken with a Lytro Illum camera.

Economic optimization of super-intensive biosecure recirculating shrimp production systems

Super-intensive, biosecure, recirculating shrimp production systems (SIBRSPS) have been developed to prevent disease outbreaks and improve profitability. To evaluate the economic viability for SIBRSPS, the Hanson-Posadas bioeconomic model was developed to conduct analysis of net present values (NPVs), internal rate of returns (IRRs), and payback periods. The objective of this study is to develop a profit maximization model, as an enhancement to the Hanson-Posadas bioeconomic model, to determine the optimal harvesting week, size, and batch number per year and evaluate the economic viability for the SIBRSPS. Biological growth and feed functions in the profit maximization model were estimated using data from experiments conducted at the Gulf Coast Research Laboratory (GCRL), Waddell Mariculture Center (WMC), and the Oceanic Institute (OI) during 2003 through 2006. Results found that Pacific white shrimp (Litopenaeus vannamei) growth and feed significantly (P < 0.01) responded to number of weeks farmed as quadratic concave function shapes in GCRL and WMC and as linear in OI. The optimal harvest week, size, and batch number per year are 12th–18th week, 18–21 g, and approximately three to four batches, respectively, depending on the location. The economic viability of SIBRSPS is questionable, with positive NPVs ($9.35, $2.27, and $0.36 million) experienced only in three (OI-14, WMC-9, and WMC-8, respectively) out of seven experiment scenarios. Profitability is dependent on economic factors including L. vannamei prices and establishment and maintenance costs of SIBRSPS as well as biological factors such as stocking density, initial puerulus (PL) size, growth rates, feed conversion rates, and survival rates.