Prismatic Elements Research Articles

Analogy of scaling law for heterogeneous interfacial fracture to explain the Gutenberg–Richter formula for earthquakes

We propose a new approach for the statistical law due to the fracture of a heterogeneous interface involving spatial correlation of disorders. The dynamic process of interfacial fracture is governed by three coupled integral equations, which further become a system of linear algebraic equations after discretizing the interface to a set of prismatic elements. By tuning parameters, this model covers the whole cases of interfacial fracture from local-load-sharing to almost equal-load-sharing, extending the classical fiber bundle models to a general form. Numerical simulations present that in all cases, the statistical frequency distribution of bursts follows a power law with the exponent in the range (1.5, 2.5), the corresponding b-value in (0.75, 2.25), which well explains the empirical Gutenberg–Richter scaling. The exponent depends on stiffness of elastic spaces, heterogeneous properties of interface, and the distribution of displacements induced by loading. Furthermore, the exponent drops temporally with the evolution of fracture, to its final value before rupture of interface, a phenomenon that may be treated as a precursor for imminent catastrophic failure.

Theory of radiation pressure on a diffractive solar sail

Solar sails propelled by radiation pressure enable space missions that cannot be achieved using chemical rockets alone. Significant in-space propulsion for missions such as a solar polar orbiter may be achieved with a sail that deviates sunlight at a large average angular direction. The momentum transfer efficiency of sunlight diffracted from a Sun-facing diffractive sail comprising periodic right prism elements is examined here. The spectrally averaged efficiency, integrated across the solar blackbody spectrum, is found to approach that of a monolithic prism when the grating period is much longer than the peak of the solar spectrum. This idealized diffraction analysis predicts a greater transverse radiation pressure force compared to an idealized reflective sail. With modern optical design and fabrication techniques, optimized diffractive solar sails may one day replace reflective sails.

High Order Wall-Modeled Large-Eddy Simulation on Mixed Unstructured Meshes

In the present study, an algebraic equilibrium wall model previously developed for hexahedral elements is extended to handle mixed meshes including prismatic, tetrahedral, and pyramidal elements in the context of a discontinuous high-order method. This extension is needed for complex geometries, for which high-order mixed elements (e.g., tetrahedral and pyramidal elements) are often necessary near solid walls to avoid meshing challenges. Various design decisions are discussed to achieve the best performance on massively parallel CPU/GPU architectures for a production-level high-order large-eddy simulation solver based on the flux reconstruction/correction procedure via reconstruction method, hpMusic. The extension to other elements is first evaluated using a benchmark channel flow problem at various Reynolds numbers. After that, flow over the NASA high-lift Common Research Model (CRM-HL) from the 4th AIAA High-lift Prediction Workshop is computed to further test the new implementation. Computational results at the third- and fourth-order accuracies are compared with experimental data.

A sequential investigation of the residual stresses and strains influence on the buckling of cold-formed thin-walled members

In this study, a specific formulation and its corresponding computational code are developed using: (i) an alternative 3D finite strain elastoplastic theory based on the multiplicative Flory decomposition, (ii) a node-surface contact algorithm to model sheet folding and (iii) a 3D prismatic solid-based finite element with high order approximation. Employing the developed code, a sequential analysis is presented that involves (i) folding an initially flat sheet in the dimensions of the studied thin-walled U-member, from which results the residual stresses of the manufacturing process, (ii) changing the boundary conditions to enable the application of linear basic loads that generate the buckling situation and (iii) solving the buckling of the cold-formed thin-walled U-member. We used literature benchmarks to show that the proposed 3D elastoplastic theory is sufficiently accurate to study the influence of residual stresses and geometric defects (residual deformations) – arising from the folding process – on the critical loads of the studied U-member.

SYSTEMS OF COORDINATE FUNCTIONS DURING THE DECOMPOSITION OF DISPLACEMENTS BY POLYNOMIALS

The formulas obtained in [10, 12] for the calculation of nodal reactions and stiffness matrix coefficients allow to use different systems of coordinate functions constructed on the basis of polynomials to represent displacements. A distinctive feature of these relations in comparison with similar ones derived when using Fourier series to representthe displacements is that the coefficients of not only diagonal, but also peripheral submatrices do not equal zero and the solution of systems of equations obtained on their basis by direct methods becomes impractical. The factors that determine the efficiency of the semi-analytical finite element method include, first of all, the task of fixing the conditions at the ends of the body and the amount of calculations due to the rate of convergence of the integration process of solving systems of equations.Of particular importance in the semi-analyticalvariant of the finite element method is the choice of the appropriate system of coordinate functions to represent the displacements along the length of the prismatic element. In deriving the relations of a universal finite element with the exception of Fourier series, the direct use of functions used in the above works to calculate prismatic bodies based on the semianalytic finite element method seems irrational, because each of them satisfies only individual cases of boundary conditions. from the standpoint of the theory of shells. In addition, among thefunctions considered, only Fourier series provide a strict separation of variables and reduce the original spatial problem to a series of two-dimensional for each contained harmonic.Based on the above research, it is concluded that mixed coordinate function systems based on Michlin polynomials allow the simplest formulation of different conditions of fixation at the ends, while ensuring a high rate of convergence of the iterative process.

Virtual elements on agglomerated finite elements to increase the critical time step in elastodynamic simulations

AbstractIn this article, we use the first‐order virtual element method (VEM) to investigate the effect of shape quality of polyhedra in the estimation of the critical time step for explicit three‐dimensional elastodynamic finite element (FE) simulations. Low‐quality FEs are common when meshing realistic complex components, and while tetrahedral meshing technology is generally robust, meshing algorithms cannot guarantee high‐quality meshes for arbitrary geometries or for non‐water‐tight computer‐aided design models. For reliable simulations on such meshes, we consider FE meshes with tetrahedral and prismatic elements that have badly shaped elements—tetrahedra with dihedral angles close to and , and slender prisms with triangular faces that have short edges—and agglomerate such “bad” elements with neighboring elements to form a larger polyhedral virtual element. On each element, the element‐eigenvalue inequality is used to estimate the critical time step. For a suite of illustrative FE meshes with being a mesh‐coordinate parameter that leads to poor mesh quality, we show that adopting VEM on the agglomerated polyhedra yield critical time steps that are insensitive as . The significant reduction in solution time on meshes with agglomerated virtual elements vis‐à‐vis tetrahedral meshes is demonstrated through explicit dynamics simulations on a tapered beam.

DETERMINATION OF STRESSES IN A CYLINDRICAL SHELL TAKING INTO ACCOUNT HOLES

Perforated elements are widely used in collectors of nuclear power plants, chemical devices, construction equipment. Perforation allows to reduce the material consumption of an expensive product, to facilitate the design and is performed for design reasons. At the same time, there is a concentration of stresses near the holes, which significantly reduces the durability of the perforated parts of the products, as well as determines the relevance of the study of the stress-strain state in the perforated parts. The paper presents an analytical description of the problem of the theory of elasticity for a perforated thick-walled cylinder in thermomechanical formulation. The problem of cylindrical bodies of rotation was considered in the axisymmetric three-dimensional formulation in displacements with the definition of stress and strain tensors. It is shown that numerical methods, in particular the finite element method, must be used to solve problems on the stress-strain state of perforated cylinders. For thick-walled perforated cylinders, it is advisable to use finite three-dimensional elements. Problems for homogeneous thick-walled and perforated cylinders were solved using three-dimensional and shell finite elements. Both four-node shell and three-dimensional eight-node prismatic finite elements were used in the numerical solution of this problem. The loading of the cylinder with holes was internal pressure. From the comparison of the results it follows that in the absence of holes it is enough to use two-dimensional shell finite elements, but in the presence of holes in perforated thick-walled shells there is a concentration of stress, so in this case it is advisable to consider the problem in three-dimensional form. It is shown in the paper that the solution of the problem taking into account the consolidated stiffness for the shell model gives an underestimated value of stresses, because it does not take into account the stress concentration on the inner surface of the shell. The coefficient of stress concentration obtained in the work due to the presence of holes was equal to two.

Additional overlapping finite elements – The pyramid and prism elements

Our objective in this paper is to present two new overlapping finite elements: a pyramid element and a prism element. These elements are useful when meshing general three-dimensional geometries in providing the possibility to transition between other overlapping and traditional elements. Both new overlapping elements are compatible with overlapping brick and tetrahedral elements, and when used as coupling elements also with the traditional finite elements. We theoretically formulate the two new overlapping finite elements, show that only the rigid body modes correspond to zero eigenvalues (hence the elements contain no spurious zero energy mode), show that the patch test is passed, and test the elements for their convergence behavior and condition numbers. We also show that the elements are quite insensitive to element distortions and revisit the imposition of displacement boundary conditions. Finally, we give some illustrative applications using the AMORE scheme in which also the new overlapping elements are employed.

A scalability study of the Ice-sheet and Sea-level System Model (ISSM, version 4.18)

Abstract. Accurately modelling the contribution of Greenland and Antarctica to sea level rise requires solving partial differential equations at a high spatial resolution. In this paper, we discuss the scaling of the Ice-sheet and Sea-level System Model (ISSM) applied to the Greenland Ice Sheet with horizontal grid resolutions varying between 10 and 0.25 km. The model setup used as benchmark problem comprises a variety of modules with different levels of complexity and computational demands. The core builds the so-called stress balance module, which uses the higher-order approximation (or Blatter–Pattyn) of the Stokes equations, including free surface and ice-front evolution as well as thermodynamics in form of an enthalpy balance, and a mesh of linear prismatic finite elements, to compute the ice flow. We develop a detailed user-oriented, yet low-overhead, performance instrumentation tailored to the requirements of Earth system models and run scaling tests up to 6144 Message Passing Interface (MPI) processes. The results show that the computation of the Greenland model scales overall well up to 3072 MPI processes but is eventually slowed down by matrix assembly, the output handling and lower-dimensional problems that employ lower numbers of unknowns per MPI process. We also discuss improvements of the scaling and identify further improvements needed for climate research. The instrumented version of ISSM thus not only identifies potential performance bottlenecks that were not present at lower core counts but also provides the capability to continually monitor the performance of ISSM code basis. This is of long-term significance as the overall performance of ISSM model depends on the subtle interplay between algorithms, their implementation, underlying libraries, compilers, runtime systems and hardware characteristics, all of which are in a constant state of flux. We believe that future large-scale high-performance computing (HPC) systems will continue to employ the MPI-based programming paradigm on the road to exascale. Our scaling study pertains to a particular modelling setup available within ISSM and does not address accelerator techniques such as the use of vector units or GPUs. However, with 6144 MPI processes, we identified issues that need to be addressed in order to improve the ability of the ISSM code base to take advantage of upcoming systems that will require scaling to even higher numbers of MPI processes.

Mixed finite elements applied to acoustic wave problems in compressible viscous fluids under piezoelectric actuation

In the present contribution, we develop a mixed finite element method capable of the coupled multi-field simulation of a viscous fluid actuated by a piezoelectric resonator. Several challenges are met with in this setting, among which are the necessity of correct interface coupling, near incompressibility of the fluid, adverse geometric dimensions of flat piezoelectric transducers and different length scales of shear and pressure wave. Assuming small deformations and velocities, we present a mixed variational formulation with consistent interface coupling conditions in (mechanic) frequency domain. Both fluid and piezoelectric solid domain are discretized using Tangential-Displacement Normal-Normal-Stress elements. These elements model not only the deformation, but add an independent tensor-valued stress approximation. The method has been rigorously proven to be free from shear locking for flat prismatic or hexahedral elements. Thus, modeling of the flat geometry of piezoelectric resonators as well as resolution of the fastly decaying shear wave are facilitated. To circumvent the problem of volume locking due to the near incompressibility of the fluid, an additional independent pressure field is introduced. We present computational results indicating the capability of the method.

Theoretical and experimental investigation of a lateral broad-crested contraction as a flow measurement device

The article looked at the possibility of making use of a lateral contraction as a flow measurement device in a rectangular open channel. The contraction under consideration consisted of two removable prismatic elements placed on either side of the walls of the channel. This configuration resulted in the creation of a throat of length L and opening width b less than the channel width B so that the contraction rate has been defined as β = b/B. This ratio actually reflects the degree of the lateral contraction while there is no vertical contraction because the device is devoid of crest height. Its longitudinal axis thus merges with that of the channel, which gives the device a self-cleaning property. Due to the changeover of the subcritical flow upstream of the device to a supercritical flow inside the throat, firmly confirmed by laboratory observations, a control section has been created in a given section at the entrance of the throat. This is the prerequisite for the correct functioning of a flow measuring device in open channels.The essential steps undertaken during the study were firstly the research for the theoretical relationship governing the flow rate Q for forthwith deriving the resulting discharge coefficient Cd equation. This aim has been successfully achieved through the simultaneous application of the momentum and the energy equations taking care to include the effect of the approach flow velocity. The obtained theoretical flow rate QTh was identified in the form of explicit functions depending both on the geometric characteristics of the device and the upstream depthh1 which is fully consistent with semi-modular devices. Likewise, the resulting discharge coefficient Cd,Th was formally identified as being exclusively dependent on the contraction rate β and this finding was predicted by the dimensional analysis.The statistical analysis of the 157 collected experimental values allowed observing an excellent agreement between the theoretical and experimental discharge coefficients. The maximum deviation calculated over the tested range of β was only 1.22%. The use of linear least-squares fitting method involving experimental and theoretical data gave the following trend line relationship Cd,Exp=0.9911Cd,Th obtained with a coefficient of determination R2=0.9906. Thereby, using the previous equation, the accuracy on the discharge coefficient Cd,Th computation has been highly improved since the maximum deviation reduced to only 0.66%.

A Systematic Approach to Constructing Preconditioners for the $hp$-Version Mass Matrix on Unstructured and Hybrid Finite Element Meshes

We develop a systematic approach to building uniform preconditioners for the mass matrix on unstructured meshes composed of higher dimensional elements using only preconditioners for lower dimensional simplices. In particular, we show that the resulting preconditioners are automatically uniform with respect to both the mesh size h and polynomial degree $p$, and that the preconditioners can be implemented efficiently by exploiting the structure of the preconditioners on the lower dimensional elements. We illustrate the approach by developing preconditioners for prismatic elements and for the challenging case of hybrid meshes of hexahedra, prisms, and tetrahedra.

A Novel Calculation Method of Hydrodynamic Pressure Based on Polyhedron SBFEM and Its Application in Nonlinear Cross-Scale CFRD-Reservoir Systems

Hydrodynamic pressure is an important factor that cannot be ignored in the seismic safety evaluation of dams. However, when the polyhedron-scaled boundary finite element method is used to simulate dams in a cross-scale dynamic analysis, polygonal surfaces often appear on the upstream face of dams, which is difficult to deal with for conventional methods of hydrodynamic pressure. In this paper, a three-dimensional calculation method of hydrodynamic pressure based on the polyhedron-scaled boundary finite element method is proposed, in which polygon (triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, etc.) semi-infinite prismatic fluid elements are constructed using the mean-value shape function. The proposed method, with a high efficiency, overcomes the limitation of conventional methods in which only quadrangle or triangle boundary faces of elements are permitted. The accuracy of the proposed method is proved to be high when considering various factors. Furthermore, combined with the polyhedron-scaled boundary finite element method for a solid dam, the proposed method for reservoir water is used to develop a nonlinear dynamic coupling method for cross-scale concrete-faced rockfill dam-reservoir systems based on the polyhedron SBFEM. The results of the numerical analysis show that when the hydrodynamic pressure is not considered, the error of rockfill dynamic acceleration and displacement could reach 15.4% and 12.7%, respectively, and the error of dynamic face slabs’ stresses could be 24.9%, which is not conducive to a reasonable seismic safety evaluation of dams.

The effect of triangular prismatic elements on the hydraulic performance of stepped spillways in the skimming flow regime: an experimental study and numerical modeling

AbstractThe stepped spillway is a cost-effective hydraulic structure to dissipate the energy of large water flow over the spillway of a dam. In this study, the focus was placed on the effects of triangular prismatic elements (TPEs) on the hydraulic performance of a stepped spillway. Nine stepped spillway models were experimentally and numerically investigated with different shapes and layouts of TPEs. To adopt a proper turbulent model, RNG k-ε and standard k-ε models were utilized. The computational model results satisfactorily simulated the complex flow over the stepped spillway of experiment cases, including velocity distributions and pressure profiles on the step surfaces. The results indicated that the installation of TPEs on stepped spillways can be an effective way to reduce cavitation effects. Installing TPEs on stepped spillways increased the energy dissipation rate up to 54%. The performance of stepped spillways was improved when TPEs were spaced more closely together. Furthermore, the relationships between the roughness coefficient (f) and the ratio of the critical depth to the step roughness (yc/k) were obtained with high accuracy using the experimental data.

A hybrid grid-based finite-element approach for three-dimensional magnetotelluric forward modeling in general anisotropic media

Magnetotelluric (MT) forward modeling often requires the consideration of the deviation generated by anisotropic structures to avoid misleading the detection results, which is implemented by the flexible unstructured finite-element (FE) method based on tetrahedrons. However, the unstructured FE method needs a large number of small elements near the sharp boundary of the electrical structure to produce sufficiently stable grids, and this leads to a sharp increase in the degrees of freedom (DoFs) of the FE system. To this end, we develop an FE approach based on a hybrid grid. It uses the stretched prismatic elements to divide the regions around the abrupt interfaces, which can significantly reduce the number of elements needed to capture the corresponding changes of the physical fields compared to using tetrahedrons. The remaining regions are still divided with tetrahedrons in this hybrid grid in order to keep the total elements needed for the entire computational region at a minimum. By showing numerical examples of the sea- and land-based electrical anisotropy models, as well as a real anisotropic inversion model, the advantages of our method are tested by evaluating the MT response curves, the number of elements, the DoFs, the computational time, and the consumed memory. The results show that the prismatic elements are suitable for discretizing the near-surface region, seawater layer, and electrical anisotropic blocks. This approach can maintain the high accuracy of the numerical solution and reduce the number of elements required for discretizing these regions and the DoFs, thus requiring less computer memory. With this hybrid grid, the computational efficiency of the FE method can be improved for MT forward modeling for a complicated model, which combined with the lower memory consumption is very suitable to implement inversion.

Finite element modeling of the processes of elastic-plastic deformation of reclamation objects of the agro-industrial complex

Annotation. To study the processes of nonlinear deformation of reclamation objects and engineering systems of the agro-industrial complex, taking into account the plastic stage of the used structural material, a finite element model was created based on a volumetric prismatic discretization element with quadrangular bases. The plastic stage of deformation of the applied structural material of the object is taken into account on the basis of the provisions of the deformation theory of plasticity. The plasticity matrix at the (j + 1)-th stage of sequential loading was compiled as a result of applying the operation of differentiating the stress tensor components accumulated over the j previous stages of sequential loading with respect to the strain tensor components. The stiffness matrix and the column of nodal forces of a prismatic discretization element with quadrangular bases at the (j + 1)-th stage of loading were obtained by minimizing the Lagrange functional. Numerical experiments on the calculation of a tubular outlet proved the adequacy of the developed finite element model and the accuracy of calculating the required strength parameters of the studied reclamation objects and other engineering systems of the agro-industrial complex, sufficient for engineering practice.

An efficient multiphysics solid shell based finite element approach for modeling thin sheet metal forming processes

The main objective of this paper is to provide a numerical approach to study thin sheet metals involved in multiphysics manufacturing processes. The proposed approach also accounts for anisotropic plastic deformations in the context of modeling of deep drawing process. In this work, different constitutive models are implemented with a prismatic solid shell element to simulate these applications. Moreover, a new element assembly technique has been developed to permit the assembly of prismatic elements in a tetrahedral element-based finite element code. This technique splits the prism into multiple tetrahedral elements in such a way that all the cross terms are accounted for. The numerical approach has been validated using multiple tests that involve different non-linearities: geometric, material and contact. Then, more complex sheet metal processes have been simulated. The performance of the solid shell element with one element through the thickness is very comparable to the results obtained with more refined mesh of a tetrahedral element. This reduction in the number of elements accelerates the simulation, especially in the deep drawing problem and the coupled magnetic-mechanical simulation used for the magnetic pulse forming process. • Solid shell finite element implementation in tetrahedral based finite element software. • Prism division algorithm into multiple overlapping tetrahedral elements. • Solid-shell element showed better performance than MINI element in modeling thin structures problems. • Incorporate anisotropic plastic behavior in the simulation of deep drawing process. • Electromagnetic pulse forming simulation using solid shell element reduces time of simulation and maintain good accuracy.

Force-controlled burnishing process for high surface integrity on additive manufactured parts

Additive manufactured components provide a new level of shape flexibility. However, this process has shortcomings in surface quality and dimensional accuracy. As a result post-processing is often necessary. In this paper, a combined finishing operation of machining shaping and force-controlled burnishing is used for prismatic form elements. Therefore, a corresponding control strategy was implemented for this on a CNC machine tool. The focus lies on the process characteristics between the pre-machined surface structures and the force-controlled machining and in the resulting surface properties. The aim of the work is the investigation and application of the process combination on additive manufactured metallic structural elements.

Calculation of the strength of objects of water management of the environment

The article discusses the objects of water management of the environment. An algorithm for forming a matrix of the stress-strain state of a prismatic finite element with a triangular cross-section in a mixed formulation is proposed to take into account the physical nonlinearity of water management objects in the form of rotation shells. The basic relations of the flow theory for arbitrary loading are used. The desired displacements and stresses of the inner point of the finite element were approximated by linear functions in terms of their nodal values. The stress-strain state matrix was formed using a functional expressing the equality of the actual work of external and internal work of external and internal forces, in which the actual work of internal forces was represented by the difference between the total and additional work of these forces.

Features of derivation of formulas for calculation of nodal reactions and coefficients of matrix of rigidity of a finite element with averaged mechanical and geometrical parameters

Currently, the most widely used finite element method for the calculation of spatial structures, significant progress in the development of which is associated with the work of domestic and foreign scientists. In Ukrainian publications the problems of theoretical substantiation of the finite element method and its connection with other methods are considered, concrete types of finite elements and their application to various problems of mechanics of a continuous environment are studied. Much attention is paid to the choice of the appropriate shape of the finite element, the type and degree of approximating functions, as well as the development of methods for deriving stiffness matrices.
 The study of prismatic bodies with constants along one of the coordinates of mechanical and geometric parameters is most appropriate to carry out on the basis of the semi-analytical method of finite elements. Its essence is a combination of finite element sampling and decomposition of displacements in the characteristic direction by a system of trigonometric coordinate functions.
 The analysis of the literature shows that the issues related to the application of the semi-analytical finite element method to the calculation of thin-walled prismatic bodies in elastic-plastic, and massive even in elastic formulations, have not been properly reflected. In addition, there are no publications in this area devoted to the development of universal prismatic finite elements that allow you to explore massive, thin-walled and combined structures.
 The direction of this study is to create on the basis of the semi-analytical method of finite elements of an effective apparatus for numerical analysis of the stress-strain state of massive and thin-walled arbitrarily loaded properties of the material and solve a number of new practically important problems. Therefore, in this work, based on the moment diagram of finite elements, formulas for calculating nodal reactions and stiffness matrix coefficients of a finite element with averaged mechanical and geometric parameters for the study of massive, thin-walled and combined structures are derived.