Order Finite Element Research Articles

Comparative analysis for five fixations of Pauwels-I by the biomechanical finite-element method

Background: Little is known about how biomechanics govern the five fixtures such as DHS, MLS, DHS + LS, LP, and HA are accepted as common therapeutic techniques. Aims and objectives: A series of numerical models for a femoral neck fracture of Pauwels-I will be constructed by innovative approach of finite element in order to determine the most optimized option in comparison with biomechanical performance. Method: Twenty sets of computer tomography scanned femora were imported onto Mimics to extract 3 D models; these specimens were transferred to Geomagic-Studio for a simulative osteotomy and kyrtograph; then, they underwent UG to fit simulative solid models; 5 sorts of fixture were then expressed by Pro-Engineer virtually. After processing with HyperMesh, all compartments (fracture model + internal implant) were assembled onto 5 systems: “Dynamic Hip Screw (DHS), Multiple Lag screw (MLS), DHS + LS, femoral Locking Plate (LP) and HemiArthroplasty (HA).” Eventually, numerical models of the finite-elemental analysis were exported to AnSys to determine the solution. Result: Four models of fixation and a simulation of HA for Pauwels-I were established, validated, and analyzed with the following findings: In term of displacement, these 5 fixtures ranged between 0.3801 and 0.7536 mm have no significant difference; in term of stress, the averages of peaks for integral assemblage are b(MLS) = 43.5766 ≈< d(LP) = 43.6657 ≈< e(Ha) = 43.6657 < c(DHS + LS) = 66.5494 < a(DHS) = 105.617 in MPa indicate that MLS, LP and HA are not significantly different, but less than DHS + LS or DHS in each. Conclusion: A fixture of MLS or LP with optional HA should be recommended to clinically optimize a Pauwels-I facture.

Enhanced Assumed Strain Methods for Implicit Gradient‐Enhanced Damage‐Plasticity

AbstractThe combination of plasticity theory and continuum damage mechanics facilitates the description of hardening and softening material behavior, accumulation of inelastic strains, and stiffness degradation due to damage. For regularizing softening material behavior in finite element simulations, the enrichment of the constitutive relations by means of a nonlocal counterpart of an internal strain‐like thermodynamic state variable, implicitly defined as the solution of a higher order PDE, is a suitable approach in order to obtain mesh‐insensitive results [1]. However, for plasticity models assuming dilatant plastic flow, such as commonly employed for cohesive‐frictional materials like concrete or rock, spurious locking phenomena due to kinematic constraints can arise in conjunction with low order finite elements. This issue was observed and discussed in [2] for simple (local) linear‐elastic perfectly plastic material models. As a remedy, specific enhanced assumed strain (EAS) methods to enrich the normal strain rate field were proposed and assessed. The present contribution demonstrates the formulation of these EAS methods combined with the fully coupled displacement‐nonlocal damage framework, and shows an application to mode II failure employing a gradient‐enhanced damage‐plasticity model for concrete. The insensitivity of the presented approach with respect to the finite element mesh is discussed.

On the construction of a triangular Mixed Finite Element based on the principle of Hellinger‐Reissner

AbstractA low order finite element based on a Hellinger‐Reissner formulation with weak enforcement of the stress symmetry is presented. The novel element gets along without any additional degrees of freedom beside the displacements and stresses. Therein, lowest order Raviart‐Thomas interpolation is utilized for the stresses and a C0 continuous polynomial interpolation of first order is adopted for the displacements. This combination leads to a very efficient and robust element formulation which will be depicted on a couple of numerical examples.

Vibration reduction of the blisk by damping hard coating and its intentional mistuning design

This paper presents a vibration reduction approach of the blisk by using of the damping hard coating and its intentional mistuning design. Firstly the amplified forced vibration responses of mistuned blisk were studied based on the reduced order finite element model (FEM) established by component mode synthesis (CMS) method, by using the actual mistuning value of a blisk specimen. Then the effects of damping hard coating on forced vibration responses of the tuned and mistuned blisk were investigated. Further, intentional mistuning design by depositing hard coating with different thickness on blades was researched, after comparing several mistuning patterns. At last, experiment research was conducted by using a blisk specimen with random mistuning. NiCrAlY coating was deposited on blades with designed thickness, in accordance with the simulation results. It reveals to be an effective approach for reducing the vibration level of mistuned blisk, judging from the observed lower vibration response level after the intentional mistuning design by hard coating. The results of this paper can help to decrease the fatigue failure induced by mistuned vibration and improve the structure integrity and security of the blisk in aero-engine.

A Schur-complement model-order-reduction technique for the finite element solution of transient elastohydrodynamic lubrication problems

The current work presents a reduced order finite element modeling framework for the solution of transient elastohydrodynamic lubrication (EHL) problems. The model order reduction technique is based on a Schur-complement method, applied to the elastic part within EHL problems. As such, it is exact and introduces no additional errors to the solution, with respect to the standard non-reduced finite element model. The technique reduces by one, the dimension of the linear elasticity part within the EHL problem. The use of the Schur-complement method however leads to a semi-dense Jacobian matrix. This is why the technique is complemented with a splitting procedure, allowing it to retrieve a standard finite-element-like sparsity pattern. In terms of computational performance, it is shown through a set of numerical experiments that the proposed reduced model offers a speed-up in computational times of the order of 15:1, compared to the equivalent full model, without any compromise on the accuracy of the solution.

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.

Coupling a NURBS contact interface with a higher order finite element discretization for contact problems using the mortar method

In this paper, a contact problem between two bodies, discretized by finite elements, is solved by adding an auxiliary NURBS layer between the bodies. The advantages of a smooth contact formulation in a NURBS approach are combined with simple mesh generation procedures for the bodies discretized with finite elements. Mesh tying conditions are used to couple the NURBS layer with the finite element discretization. The NURBS layer is the master side for contact and mesh tying. Mesh tying is enforced either using pointwise or mortar type approaches. Frictionless 2D and 3D contact problems are considered using small deformations. The contact problem is discretized with the mortar method and a penalty approach is used to enforce the contact constraints. A robust element-based quadrature is applied for mortar tying and contact discretizations, thus avoiding computationally expensive segmentation.

Reduced order finite element formulations for vibration reduction using piezoelectric shunt damping

The present work proposes an original reduced order model for prediction of passive reduction of structural vibration by means of shunted piezoelectric patches. The problem consists of an elastic structure with surface-mounted piezoelectric patches. The piezoelectric elements are connected to a resonant shunt circuits in order to damp specific resonant frequencies of the structure. An efficient electromechanical finite element formulation for the dynamic analysis of the problem is first presented. The classical modal superposition techniques using the system eigenvectors with all patches short-circuited or open-circuited are then recalled. An advanced reduced order models using two new modal projection bases able to solve the problem at lower cost are developed: (i) the combined basis formed by both the short-circuited and open-circuited modes, and (ii) the coupled basis formed by the electromechanical modes that take into account the effect of the inductances of the electrical shunt circuits. Various numerical and experimental results are presented in order to validate and illustrate the efficiency of the proposed new finite element reduced order formulations.

GPU‐based Polynomial Finite Element Matrix Assembly for Simplex Meshes

AbstractIn this paper, we present a matrix assembly technique for arbitrary polynomial order finite element simulations on simplex meshes for graphics processing units (GPU). Compared to the current state of the art in GPU‐based matrix assembly, we avoid the need for an intermediate sparse matrix and perform assembly directly into the final, GPU‐optimized data structure. Thereby, we avoid the resulting 180% to 600% memory overhead, depending on polynomial order, and associated allocation time, while simplifying the assembly code and using a more compact mesh representation. We compare our method with existing algorithms and demonstrate significant speedups.

The numerical study of a microscale model for lithium-ion batteries

We consider a thermodynamic consistent microscale model from for the transport processes of lithium ion concentration and charge in lithium ion batteries. A fully convergent finite element discretisation in space of arbitrary order with uniform time stepping is discussed for this strongly nonlinear coupled system of nonlinear PDEs. We analyse the error for higher order finite element methods on a representative geometry, which shows already the difficulties when dealing with a realistic microstructure. Moreover we derive explicit analytical formulas for the exact solutions for the elliptic subproblem and the complete time-dependent problem for a non-trivial geometry. These results show the need for appropriate boundary approximations as well as adaptive refinement strategies. In addition we incorporate a two-phase intercalation model for the electrode. This moving boundary model, also called Stefan problem, has also been implemented. Simulations of our battery model including the two-phase model will be shown.

Optimal estimates on stabilized finite volume methods for the incompressible Navier–Stokes model in three dimensions

Optimal estimates on stabilized finite volume methods for the three dimensional Navier–Stokes model are investigated and developed in this paper. Based on the global existence theorem [23], we first prove the global bound for the velocity in the H1‐norm in time of a solution for suitably small data, and uniqueness of a suitably small solution by contradiction. Then, a full set of estimates is then obtained by some classical Galerkin techniques based on the relationship between finite element methods and finite volume methods approximated by the lower order finite elements for the three dimensional Navier–Stokes model.

The T-stress along a 3-D straight crack

The T-stress in a three dimensional elastic domain containing a straight crack is considered. First the complete asymptotic expansion in the vicinity of the crack front, including the T-stress, is presented explicitly. Having obtained the asymptotic expansion, we compute the dual solution associated with the T-stress, and prove it must contain logarithmic terms. This dual solution is used to extend the quasi dual function method (QDFM) for the extraction of the T-stress from finite element solutions and express it as a function along the crack front.Since the stress field tends to infinity at the crack front whereas the T-stress remains constant, its extraction from finite element solutions, containing numerical errors, is a non-trivial task. We herein develop accurate and efficient methods for extracting the T-stress function along the crack edge by the QDFM. Numerical examples are provided for the computation of the T-stresses from conventional and high order finite element methods.

A partition of unity finite element method for computational diffusion MRI

The Bloch–Torrey equation describes the evolution of the spin (usually water proton) magnetization under the influence of applied magnetic field gradients and is commonly used in numerical simulations for diffusion MRI and NMR. Microscopic heterogeneity inside the imaging voxel is modeled by interfaces inside the simulation domain, where a discontinuity in the magnetization across the interfaces is produced via a permeability coefficient on the interfaces. To avoid having to simulate on a computational domain that is the size of an entire imaging voxel, which is often much larger than the scale of the microscopic heterogeneity as well as the mean spin diffusion displacement, smaller representative volumes of the imaging medium can be used as the simulation domain. In this case, the exterior boundaries of a representative volume either must be far away from the initial positions of the spins or suitable boundary conditions must be found to allow the movement of spins across these exterior boundaries.Many approaches have been taken to solve the Bloch–Torrey equation but an efficient high performance computing framework is still missing. In this paper, we present formulations of the interface as well as the exterior boundary conditions that are computationally efficient and suitable for arbitrary order finite elements and parallelization. In particular, the formulations are based on the partition of unity concept which allows for a discontinuous solution across interfaces conforming with the mesh with weak enforcement of real (in the case of interior interfaces) and artificial (in the case of exterior boundaries) permeability conditions as well as an operator splitting for the exterior boundary conditions. The method is straightforward to implement and it is available in FEniCS for moderate-scale simulations and in FEniCS-HPC for large-scale simulations. The order of accuracy of the resulting method is validated in numerical tests and a good scalability is shown for the parallel implementation. We show that the simulated dMRI signals offer good approximations to reference signals in cases where the latter are available and we performed simulations for a realistic model of a neuron to show that the method can be used for complex geometries.

Geometric element parameterization and parametric model order reduction in finite element based shape optimization

This contribution proposes a new approach to derive geometrically parameterized, reduced order finite element models. An element formulation for geometrically parameterized finite elements is suggested. The parameterized elements are used to derive models with a parameterized geometry where the parameterized system matrices are expressed in an affine representation. Parametric model order reduction can then be efficiently used to reduce the full order parameterized model to a reduced order parameterized model. The approach shows two beneficial features. First, design studies and shape optimizations can be conducted with parameterized reduced order model of much lower dimension compared to the parameterized, full order model. Second, it is possible to compute sensitivities analytically, and therefore, to avoid the computation of finite differences gradients. The approach is illustrated with two numerical examples. The first example includes a detailed error analysis. The second example is a shape optimization example of an adaptive structure.

Assessment of delaminated smart composite laminates via system identification and supervised learning

This paper proposes the synergetic integration of system identification and artificial intelligence for the detection and assessment of delamination damages in smart composite laminates. An electromechanically coupled mathematical model is developed for the healthy and delaminated smart composite laminates on the basis of improved layerwise theory, higher order electric potential field and finite element method. A discriminative feature space is constructed for the healthy and delaminated structures via system identification from their structural vibration responses. The discriminative features are used for the training and cross-validation of various supervised machine learning classifiers and an optimal classifier is identified. The optimal classifier is employed to make predictions on unseen test delamination cases, and its predictions are validated via a dimensionality reduction tool. The obtained results show that the proposed technique could be employed as a reliable tool for nondestructive evaluation of smart composite laminates.

Adaptive stabilized finite volume method and convergence analysis for the Oseen equations

In this paper, based on the pressure project method, we consider an adaptive stabilized finite volume method for the Oseen equations with the lowest equal order finite element pair. Firstly, we develop the discrete forms in both finite element and finite volume methods, and establish the existence and uniqueness of numerical solutions by establishing the equivalence of linear terms in finite element and finite volume methods. Secondly, a residual type a posteriori error estimator is designed, and the computable global upper and local lower bounds between the exact solutions and the finite volume solutions are established. Thirdly, a discrete local lower bound between two successive finite volume solutions is obtained, convergence analysis of the adaptive stabilized finite volume method is also performed. Finally, some numerical results are presented to verify the performances of the developed error estimators and confirm the established theoretical findings.

Numerical Study of Frequency and Deflection Responses of Natural Fiber (Luffa) Reinforced Polymer Composite and Experimental Validation

ABSTRACTThe natural fiber (Luffa cylindrica fiber) reinforced epoxy composite has been fabricated and their structural responses (frequency and deflection) have been computed experimentally and numerically first time using the corresponding experimental elastic properties. The numerical responses are obtained with the help of an in-house MATLAB code developed based on the higher order finite element (FE) model. The completeness of the model has been examined by comparing the current FE solutions (frequency and the central deflection) with the reference as well as in-house experimental data. The effect of fiber volume fractions on the elastic properties is verified for four different weight percentage of treated Luffa fiber (0%, 3.2%, 6.4%, and 9.6%). Finally, the inclusive behavior of the current higher order FE model and the corresponding influence of the significant design parameter of Luffa fiber-reinforced composite structure have been debated by solving different numerical examples.

Higher order FEM for the obstacle problem of the [formula omitted]-Laplacian—A variational inequality approach

We consider higher order finite element discretizations of a nonlinear variational inequality formulation arising from an obstacle problem with the p-Laplacian differential operator for p∈(1,∞). We prove an a priori error estimate and convergence rates with respect to the mesh size h and in the polynomial degree q under assumed regularity. Moreover, we derive a general a posteriori error estimate which is valid for any uniformly bounded sequence of finite element functions. All our results contain the known results for the linear case of p=2. We present numerical results on the improved convergence rates of adaptive schemes (mesh size adaptivity with and without polynomial degree adaptation) for the singular case of p=1.5 and for the degenerated case of p=3.

Positive scalar curvature and low-degree group homology

Let $\Gamma$ be a discrete group. Assuming rational injectivity of the Baum-Connes assembly map, we provide new lower bounds on the rank of the positive scalar curvature bordism group and the relative group in Stolz' positive scalar curvature sequence for $\mathrm{B} \Gamma$. The lower bounds are formulated in terms of the part of degree up to $2$ in the group homology of $\Gamma$ with coefficients in the $\mathbb{C}\Gamma$-module generated by finite order elements. Our results use and extend work of Botvinnik and Gilkey which treated the case of finite groups. Further crucial ingredients are a real counterpart to the delocalized equivariant Chern character and Matthey's work on explicitly inverting this Chern character in low homological degrees.

Earthquake analysis of thick plates resting on elastic foundation with second order finite element

Earthquake analysis of thick plates resting on elastic foundation with second order finite element