General Finite Element Method Research Articles

The Composite Grid Method for Singular Problems of Partial Differential Equations

Partial differential equations are crucial in scientific computing, and this paper will consider some of the problems of partial differential equation singularities. The Composite Mesh Method (CGM) is a new and improved numerical method for solving partial differential equations based on existing numerical methods for finite elements. The method has two meshes over the entire domain—a coarse and a fine set. The two sets of meshes generated by Mesh3 are separate in their respective regions and do not nest or interact. This method improves the accuracy of solving the numerical solution of partial differential equations. This paper discusses the CGM method based on the Finite Element Program Generator (FEPG) and uses it to simulate several singular problems. The numerical simulation results show that the proposed method can obtain more satisfactory simulation results for global problems and use a smaller number of computational generations than the general finite element method.

The polygonal finite element method for solving heat conduction problems

In this paper, a general polygonal finite element method (FEM-polygon) based on the Wachspress shape function is used to solve two-dimensional heat conduction problems. The finite element variational form of the heat conduction problem is first established based on the weighted residual method. Then the formulations of the thermal stiffness matrix, thermal damping matrix, and nodal heat vector are derived using the Wachspress coordinates. The global stiffness matrix is further assembled and the well-behaved algebraic equations of the thermal system are established. Finally, intensive numerical tests are conducted to verify the super convergence in temperature and the high precision in equivalent energy and temperature gradient of the present FEM-polygon for 2D heat conduction problems. Besides, the proposed model also performs excellently for traditional triangular and quadrilateral elements.

Microstructure modification and coercivity enhancement of sintered NdFeB magnet by vacuum diffusion 70Tb–20Cu–10Ga (at%) alloys

In this work, the eutectic Tb70Cu20Ga10 alloy was applied to high magnetic performance NdFeB magnet. The coercivity of magnets effectively improved of 60.61% without reduction in maximum energy product and remanence after diffusion. The magnetic vector potential of magnet was analyzed by general finite element method. The diffusion mechanism of eutectic had been discussed at various diffusion depth. The Tb diffuses into the magnet formation continuous Tb-rich shells and thin grain boundary phase, which strongly enhance coercivity. The Cu, Ga and O enrich into grain boundary form thin RE2(Fe, CuGaO)3 phases, which increase diffusion channel and recover microcrack near surface of magnet. This Tb70Cu20Ga10 diffusion alloys has a promising potential to develop high magnetic performance and low cost sintered NdFeB magnet.

A prediction method for the screening current induced field in HTS magnets based on time series models

Due to their special electromagnetic properties, high temperature superconducting (HTS) conductors have become a potential solution for ultra-high field magnet and energy storage applications. However, the screening current induced field (SCIF) has been demonstrated to be the main limitation of high field HTS magnets in actual applications. Based on time series models, this paper presents a prediction method of SCIF to support the design and application of HTS magnets. First, we analyze the data characteristics of the SCIF hysteresis loop. The simulated dataset is prepared for two typical magnet structures: single pancake and solenoid. Then, time series models are proposed for the SCIF prediction. Through intuitive analysis and evaluation metrics, the training performance of time series models is confirmed. After a discussion of hyper-parameters and dimension reduction, the optimized prediction performance is obtained for the SCIF hysteresis loop. In conjunction with the iterative prediction mode, we finally achieve a feasible and effective prediction method of SCIF for HTS magnets. This will provide a tool and research strategy to support the general finite element method.

Super-high coercivity NdFeB magnet fabricated with double Tb-rich/lean shells by double alloy method and grain boundary diffusion

In this work, the high performance NdFeB magnet were fabricated by double alloy method and grain boundary diffusion Tb70Cu20Ga10 alloys. The maximum energy product (BH)max and coercivity Hcj of the magnet are 45.33 MGOe and 35.46 kOe, respectively. The reversible temperature coefficients of coercivity of the diffusion magnets improve from − 0.456%/°C to − 0.412%/°C at the range of 22 °C −180 °C. The magnetostatic loss of NdFeB magnet was analyzed in high temperature applications by general finite element method (FEM). The RE-(Fe,CuGaO) grain boundary phases and double Tb-rich/lean shells improve the utilization rate of heavy rare earths and coercivity of magnets after diffusion. The magnetization reversal mechanism of double core-shell microstructures had been discussed. This work may shed light on developing coercivity above 30 kOe of NdFeB magnet with low heavy rare earth through modified microstructure.

Novel bi-layer beam elements for elastic fracture analysis of delaminated composite beams

A general finite element method is proposed for linear elastic fracture analysis of delaminated bilayer composite beams. An interface deformable bilayer beam model is adopted in modeling the virgin portions of bilayer beams, and it is capable of guaranteeing the continuity condition of internal forces at the tip of delamination because the bilayers are modeled as two separate shear deformable sub-beams and both the interface normal and shear compliances are introduced to account for interface deformation. The stiffness matrices of bilayer beams using the linear and quadratic interpolation polynomials are both derived using the principle of minimum potential energy. Then, a finite element program is developed by assembling the element stiffness matrices of virgin and delaminated portions of bilayer beams and applying the corresponding equivalent nodal forces. Two typical fracture specimens (i.e., single leg bending and end-notched flexure bilayer beams) are analyzed using the proposed bilayer beam elements. The computed displacement and energy release rate are compared well with those of analytical solutions, and the mode components of energy release rate for single leg bending specimen are also obtained using the virtual crack closure technique. It demonstrates that the proposed novel bilayer beam elements are capable of effectively and efficiently predicting the displacement and energy release rate (including its mode components) of delaminated bilayer beams. The developed bilayer beam element is general in nature and accurate and efficient in particular, because it can achieve the same accuracy as the high order beam theory and be easily implemented in the existing commercial finite element software.

A new general finite element method for predicting shearing and piercing

A new general finite element method for predicting shearing and piercing

Damage detection in structural components using free vibration analysis

The primary aim of vibration analysis in Structural Engineering is to detect damages within a structural component so that the damage quantification can be done in order to repair it to increase its serviceability. Dynamic characteristics of a damaged and healthy component differ because of change in stiffness/mass and hence can be used for the detection of damage. In the present paper, the Eigenvalues/Eigenvectors of a uniform cantilever healthy beam are determined analytically using Euler-Bernoulli’s approach and by the general finite element method approach (ANSYSTM). Free Vibration analysis is carried out by inducing the damage in the form of stiffness reduction & modal characteristics are extracted. The damage quantification is done by comparing Eigen values and Eigen vectors from the modal analysis.

A general finite element method: Extension of variational analysis for nonlinear heat conduction with temperature-dependent properties and boundary conditions, and its implementation as local refinement

In simulation of heat conduction with temperature-independent physical properties and boundary conditions (BCs), Galerkin residual analysis and variational analysis yield equivalent finite element method (FEM), the conventional FEM. However, if the properties and BCs are temperature-dependent, it is discovered that their derivatives further induce nonlinearity of FEM which consequently generates divergence between the two analyses. A general FEM, extension of variational analysis, is derived as general form of conventional FEM modeling nonlinear heat conduction. Numerical examples demonstrate that the general FEM produces results with considerably higher accuracy and stability and also possesses higher performances on conforming with both two analyses. Since general FEM degenerates to conventional FEM if derivatives are of small-amplitude or zero and its direct implementation to the entire domain is costive, general FEM is alternatively utilized as local refinement of governing equation only to points with significant derivatives. The strategy of local refinement is optimized to enhance efficiency.

On Thermodynamically Compatible Finite Volume Methods and Path-Conservative ADER Discontinuous Galerkin Schemes for Turbulent Shallow Water Flows

In this paper we propose a new reformulation of the first order hyperbolic model for unsteady turbulent shallow water flows recently proposed in Gavrilyuk et al. (J Comput Phys 366:252–280, 2018). The novelty of the formulation forwarded here is the use of a new evolution variable that guarantees the trace of the discrete Reynolds stress tensor to be always non-negative. The mathematical model is particularly challenging because one important subset of evolution equations is nonconservative and the nonconservative products also act across genuinely nonlinear fields. Therefore, in this paper we first consider a thermodynamically compatible viscous extension of the model that is necessary to define a proper vanishing viscosity limit of the inviscid model and that is absolutely fundamental for the subsequent construction of a thermodynamically compatible numerical scheme. We then introduce two different, but related, families of numerical methods for its solution. The first scheme is a provably thermodynamically compatible semi-discrete finite volume scheme that makes direct use of the Godunov form of the equations and can therefore be called a discrete Godunov formalism. The new method mimics the underlying continuous viscous system exactly at the semi-discrete level and is thus consistent with the conservation of total energy, with the entropy inequality and with the vanishing viscosity limit of the model. The second scheme is a general purpose high order path-conservative ADER discontinuous Galerkin finite element method with a posteriori subcell finite volume limiter that can be applied to the inviscid as well as to the viscous form of the model. Both schemes have in common that they make use of path integrals to define the jump terms at the element interfaces. The different numerical methods are applied to the inviscid system and are compared with each other and with the scheme proposed in Gavrilyuk et al. (2018) on the example of three Riemann problems. Moreover, we make the comparison with a fully resolved solution of the underlying viscous system with small viscosity parameter (vanishing viscosity limit). In all cases an excellent agreement between the different schemes is achieved. We furthermore show numerical convergence rates of ADER-DG schemes up to sixth order in space and time and also present two challenging test problems for the model where we also compare with available experimental data.

Development of high-performance sports machine and elucidation of Two-seam ball gyroball

Sports engineering uses mathematics and physics to develop solutions to sporting problems. It involves the design and manufacture of sports equipment and facilities that can help maximise athletic performance. In order to develop fit-for-purpose products, it is necessary to consider the motions and movements of athletes. Professor Shinobu Sakai, Production Systems Engineering and Sciences, Komatsu University, Japan, is combining AI and neural networks (NN) to develop multifunctional and high-performance sports equipment and training machines that can benefit athletes and people across the globe who play sports. Believing that machines are subordinate to human beings, a key goal for Sakai is to augment the performance of sports players through the development of tools and machines. Some of the state-of-the-art tools and methodologies he uses in his work are high-resolution sensors, laser-speed sensors, MATLAB and general finite element method analysis software (ANSYS/LS-DYNA). He has developed a shuttlecock launching machine for badminton that can launch at high speed, a high-performance baseball pitching machine that is able to generate an efficient and objective gyro spin and has a high throwing performance and a pitching control method that uses AI.

Towards rigorous boundary value level sensitivity analyses using FEM

The increased complexity of contemporary constitutive models for soils requires a rigorous method to evaluate the effect of the large number of model parameters on the results. Ideally, the interaction effects between the individual parameters should also be quantified. This is achieved by combining a state-of-the-art global sensitivity method with a general purpose Finite Element Method (FEM) for geotechnics. The method is tested for the non-trivial example of coupled hydro-mechanical response of clay in oedometric compression. The results indicate that proposed method for rigorous sensitivity analysis provides a feasible, yet more powerful, alternative to the method commonly used by engineers, i.e. the sequences of one-factor-at-a-time (OFAT) trails.

Arbitrary Lagrangian–Eulerian finite element method for curved and deforming surfaces: I. General theory and application to fluid interfaces

An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.

A generalized poroelastic model using FEniCS with insights into the Noordbergum effect

Understanding the coupled behavior of fluid flow and solid deformation in porous media is often a critical aspect of geoscientific investigations, including studies of injection-induced seismicity, geological carbon sequestration, or surface deformation from pumping operations. In 1941, Biot first outlined the standard poroelastic formulation that accounted for this coupling within an isotropic and elastic porous medium. This fully coupled system of partial differential equations typically requires the use of numerical modeling for any practical purposes, necessitating either lengthy code development or the use of a complex, prebuilt model that often lacks flexibility. However, due to recent advances in automated approaches to solving systems of differential equations, problems of poroelasticity can now be easily handled in a streamlined yet flexible manner. Here, we present a poroelastic model built within the framework of FEniCS – an open source, general purpose finite element method software – which solves the system monolithically and can produce a continuous and mass-conserving solution for specific discharge. We present both a linear model and a generalized, nonlinear model. The nonlinear model allows for variable fluid density and employs a fluid pressure and volumetric strain dependent porosity relationship. The behavior of the linear model is verified with common benchmark problems, whereas results from the nonlinear model are compared to assess the impact of including its generalizations. Finally, the model is applied to a two-dimensional problem of fluid extraction meant to replicate the well-known Noordbergum effect (in which the fluid pressure in layers adjacent to the pumped layer temporarily increases during pumping). This is often referred to as ”reverse water fluctuation”. This showcases not only the flexibility of the model but also its ability to simulate numerically challenging scenarios. The results from this simulation suggest a novel explanation of the physical mechanism generating the Noordbergum effect: strain gradients.

Constraint conditional finite element method for off-axial interfacial sliding of fiber reinforced composite

The main toughening mechanism of Ceramics Matrix Composite (CMC) is the fiber/matrix interfacial debonding and sliding. To simulate of the contact problem (such as interfacial sliding), penalty method etc have been used in general finite element method. But, these method need the repeated calculations, thus the CPU time is long and the accuracy depends on the number of repetitions. On the other hand, in actual CMCs, the fibers are not necessarily oriented along the loading direction, and the fiber diameter also fluctuates along the axis. Therefore, in this study, Constraint Conditional Finite Element Method (CC-FEM) was formulated to analyze the “off-axis interfacial sliding” with high precision and in a short time without repeated calculations. In CC-FEM, the equality of nodal displacements at the interface and the equilibrium of contact forces are assumed as constraint conditions, in which Coulomb's friction law and equivalent friction coefficient were taken into account. In addition, the validity was verified especially for “off-axis interfacial sliding” by comparing with general-purpose finite element software ANSYS. In the both cases of on- and off-axial interfacial sliding, the resultant stress distributions of the fiber and matrix agreed well with those of ANSYS. As compared to the case of on-axial interfacial sliding, the matrix stress recovered more steeply because of the higher equivalent frictional coefficient.

Root iterative method for static performance analysis of aerostatic thrust bearings with multiple pocketed orifice-type restrictors based on ANSYS

PurposeThe purpose of this paper is to present a novel numerical approach to analyze the static performance of aerostatic thrust bearings by adopting a general finite element method calculation program.Design/methodology/approachThe characteristics of a gas film are described by the Reynolds equation and the pressure distribution is solved using the finite element method. A root iterative method is proposed to meet the requirement of the mass-conservation law because multiple pocketed orifice-type restrictors are treated as a series of special boundary conditions.FindingsThe static performance of a rotary table using aerostatic thrust bearings, including load carrying capacity and stiffness, can be predicted by the method; moreover, it can be further confirmed through experiments on the designed rotary table.Originality/valueThe method combining the finite element and root iterative methods is highly accurate and has a low time-cost for analyzing aerostatic thrust bearings with multiple pocketed orifice-type restrictors.

Novel quadtree algorithm for adaptive analysis based on cell-based smoothed finite element method

For finite element analysis of structures with irregular geometry, unsatisfactory meshing quality could be the major concern. Adaptive meshing technique is usually applied to modulate the mesh to approach the geometry. The stress solution from triangular elements is understood less accurate than that of quadrilateral elements. Therefore, adaptive technique for quadrilateral elements is desirable for more proficient analysis of complex geometry. This paper develops a novel adaptive technique that combines a newly developed quadtree algorithm with the cell-based smoothed finite element method (CS-FEM) for automatic mesh adaptation using quadrilateral elements. Since the quadtree algorithm generates a grid with different sizes of quadrilaterals, a number of new CS-FEM elements of n-sided polygon are concomitantly constituted. In the adaptation process, the energy error norm and the geometrical feature are applied as the calibrations for the adaptive refinement. The elements on curved boundaries are either cut or merged with the surrounding elements automatically. It is found that the present new quadtree algorithm works very effectively with the CS-FEM formulation in view of stability and convergence. The results by the current S-FEM are found generally more accurate than those of the general finite element method (FEM) counterpart, especially in terms of strain energy solutions.

Influence of wear on hot banding migration of sealing ring using FEM

Hot banding is a typical thermoelastic instability phenomenon that occurs on the sealing surface of the sealing ring in a transmission. The presence of hot banding on sealing contact surfaces causes local wear. The wear of a sealing ring results in the loss of supply pressure and ultimately causes transmission failure. Local wear and hot banding migration also affect each other. This paper has discussed the influence of the wear of the sealing interface on hot banding migration using the general finite element method. A wear simulation procedure, based on Archard's wear law, was used to calculate the wear of the sealing ring, and the thermoelastic effects were also included in the simulation model. An iterative cyclic updating method was performed using the general finite element analysis software. The results of the simulation showed how wear influenced the migration of the hot banding. The worn sealing surfaces lead to more complex migration states of the hot banding, including division and combination of the hot banding. The proposed simulation method and the procedure that was used are suitable for trend analysis of seal design.

Numerical Solution of Nonlinear Schrödinger Equation with Neumann Boundary Conditions Using Quintic B-Spline Galerkin Method

This paper is concerned with the numerical solution of the nonlinear Schrödinger (NLS) equation with Neumann boundary conditions by quintic B-spline Galerkin finite element method as the shape and weight functions over the finite domain. The Galerkin B-spline method is more efficient and simpler than the general Galerkin finite element method. For the Galerkin B-spline method, the Crank Nicolson and finite difference schemes are applied for nodal parameters and for time integration. Two numerical problems are discussed to demonstrate the accuracy and feasibility of the proposed method. The error norms L 2 , L ∞ and conservation laws I 1 , I 2 are calculated to check the accuracy and feasibility of the method. The results of the scheme are compared with previously obtained approximate solutions and are found to be in good agreement.

Highly accurate acoustic scattering: Isogeometric Analysis coupled with local high order Farfield Expansion ABC

This work is concerned with a unique combination of high order local absorbing boundary conditions (ABC) with a general curvilinear Finite Element Method (FEM) and its implementation in Isogeometric Analysis (IGA) for time-harmonic acoustic waves. The ABC employed were recently devised by Villamizar et al. (2017) . They are derived from exact Farfield Expansions representations of the outgoing waves in the exterior of the regions enclosed by the artificial boundary. As a consequence, the error due to the ABC on the artificial boundary can be reduced conveniently such that the dominant error comes from the volume discretization method used in the interior of the computational domain. Reciprocally, the error in the interior can be made as small as the error at the artificial boundary by appropriate implementation of p- and h-refinement. We apply this novel method to cylindrical, spherical and arbitrary shape scatterers including a prototype submarine. Our numerical results exhibit spectral-like approximation and high order convergence rate. Additionally, they show that the proposed method can reduce both the pollution and artificial boundary errors to negligible levels even in very low- and high-frequency regimes with rather coarse discretization densities in the IGA. As a result, we have developed a highly accurate computational platform to numerically solve time-harmonic acoustic wave scattering in two- and three-dimensions.