Smooth Strains Research Articles

GMDIC: a digital image correlation measurement method based on global matching for large deformation displacement fields

The digital image correlation method is a non-contact optical measurement method, which has the advantages of full-field measurement, simple operation, and high measurement accuracy. The traditional DIC method can accurately measure displacement and strain fields, but there are still many limitations. (i) In the measurement of large displacement deformations, the calculation accuracy of the displacement field and strain field needs to be improved due to the unreasonable setting of parameters such as subset size and step size. (ii) It is difficult to avoid under-matching or over-matching when reconstructing smooth displacement or strain fields. (iii) When processing large-scale image data, the computational complexity will be very high, resulting in slow processing speeds. In recent years, deep-learning-based DIC has shown promising capabilities in addressing the aforementioned issues. We propose a new, to the best of our knowledge, DIC method based on deep learning, which is designed for measuring displacement fields of speckle images in complex large deformations. The network combines the multi-head attention Swin-Transformer and the high-efficient channel attention module ECA and adds positional information to the features to enhance feature representation capabilities. To train the model, we constructed a displacement field dataset that conformed to the real situation and contained various types of speckle images and complex deformations. The measurement results indicate that our model achieves consistent displacement prediction accuracy with traditional DIC methods in practical experiments. Moreover, our model outperforms traditional DIC methods in cases of large displacement scenarios.

Enhanced meshfree method with nodal integration for analysis of functionally graded material sandwich curved shells

This paper presents a nodal integration technique, the sub-domain stabilized conforming integration (SSCI), for the meshfree radial point interpolation method (RPIM) applied to the static and modal analysis of functionally graded material (FGM) sandwich curved shells. FGM sandwich shells with different kinds of core and face sheets are considered in this work while the interested curved shell is formulated by the first-order shear deformation theory. The numerical integration technique to compute the stiffness and mass matrices in the equilibrium equation is the SSCI, which is a stabilized nodal integration with strain smoothing to preserve the accuracy and stability of the numerical results. The RPIM shape functions are utilized in this study for interpolating both the field variables and the geometry of the curved shell due to their ability to satisfy the Kronecker delta property, a rare advantage among meshfree methods. The static and modal analysis of different geometry curved shells with various sandwich FGMs are conducted. Through several numerical examples, the accuracy and efficiency of the SSCI technique in the meshfree RPIM are demonstrated and discussed.

A MITC3+ element improved by edge-based smoothed strains for free vibration and buckling analyses of porous plates based on the first-order shear deformation theory

This paper presents the free vibration and critical buckling analyses of porous plates based on the first-ordershear deformation theory by the proposed three-node triangular element. The bending strain fields of the suggested element are enriched by the bubble node located at the centroid of the triangular element. The shear locking phenomenon is eliminated by the independent interpolations of the transverse shear strains following the MITC3+ technique. The edge-based smoothed (ES) strain method is employed to improve the in-plane strain fields. The influence of the porosity distributions, length-to-thickness ratios, porous coefficient, and boundary conditions on the free vibration and critical buckling load of the porous plates are evaluated through several numerical examples by the proposed element, namely ES-MITC3+ element. The obtained results are compared with other references to perform the efficiency of the proposed element.

The autophagy-targeting compound V46 enhances antimicrobial responses to Mycobacteroides abscessus by activating transcription factor EB

Mycobacteroides abscessus (Mabc) is a rapidly growing nontuberculous mycobacterium that poses a considerable challenge as a multidrug-resistant pathogen causing chronic human infection. Effective therapeutics that enhance protective immune responses to Mabc are urgently needed. This study introduces trans-3,5,4′-trimethoxystilbene (V46), a novel resveratrol analogue with autophagy-activating properties and antimicrobial activity against Mabc infection, including multidrug-resistant strains. Among the resveratrol analogues tested, V46 significantly inhibited the growth of both rough and smooth Mabc strains, including multidrug-resistant strains, in macrophages and in the lungs of mice infected with Mabc. Additionally, V46 substantially reduced Mabc-induced levels of pro-inflammatory cytokines and chemokines in both macrophages and during in vivo infection. Mechanistic analysis showed that V46 suppressed the activation of the protein kinase B/Akt-mammalian target of rapamycin signaling pathway and enhanced adenosine monophosphate-activated protein kinase signaling in Mabc-infected cells. Notably, V46 activated autophagy and the nuclear translocation of transcription factor EB, which is crucial for antimicrobial host defenses against Mabc. Furthermore, V46 upregulated genes associated with autophagy and lysosomal biogenesis in Mabc-infected bone marrow-derived macrophages. The combination of V46 and rifabutin exerted a synergistic antimicrobial effect. These findings identify V46 as a candidate host-directed therapeutic for Mabc infection that activates autophagy and lysosomal function via transcription factor EB.

The diversity of clinical Mycobacterium abscessus isolates in morphology, glycopeptidolipids and infection rates in a macrophage model.

Introduction. Mycobacterium abscessus (MABS) is a pathogenic bacterium that can cause severe lung infections, particularly in individuals with cystic fibrosis. MABS colonies can exhibit either a smooth (S) or rough (R) morphotype, influenced by the presence or absence of glycopeptidolipids (GPLs) on their surface, respectively. Despite the clinical significance of these morphotypes, the relationship between GPL levels, morphotype and the pathogenesis of MABS infections remains poorly understood.Gap statement. The mechanisms and implications of GPL production and morphotypes in clinical MABS infections are unclear. There is a gap in understanding their correlation with infectivity and pathogenicity, particularly in patients with underlying lung disease.Aim. This study aimed to investigate the correlation between MABS morphology, GPL and infectivity by analysing strains from cystic fibrosis patients' sputum samples.Methodology. MABS was isolated from patient sputum samples and categorized by morphotype, GPL profile and replication rate in macrophages. A high-content ex vivo infection model using THP-1 cells assessed the infectivity of both clinical and laboratory strains.Results. Our findings revealed that around 50 % of isolates displayed mixed morphologies. GPL analysis confirmed a consistent relationship between GPL content and morphotype that was only found in smooth isolates. Across morphotype groups, no differences were observed in vitro, yet clinical R strains were observed to replicate at higher levels in the THP-1 infection model. Moreover, the proportion of infected macrophages was notably higher among clinical R strains compared to their S counterparts at 72 h post-infection. Clinical variants also infected THP-1 cells at significantly higher rates compared to laboratory strains, highlighting the limited translatability of lab strain infection data to clinical contexts.Conclusion. Our study confirmed the general correlation between morphotype and GPL levels in smooth strains yet unveiled more variability within morphotype groups than previously recognized, particularly during intracellular infection. As the R morphotype is the highest clinical concern, these findings contribute to the expanding knowledge base surrounding MABS infections, offering insights that can steer diagnostic methodologies and treatment approaches.

Characterization of a Salmonella enterica serovar Typhimurium lineage with rough colony morphology and multidrug resistance

Salmonella enterica serovar Typhimurium (S. Typhimurium) is a major cause of salmonellosis, and the emergence of multidrug-resistant pathovariants has become a growing concern. Here, we investigate a distinct rough colony variant exhibiting a strong biofilm-forming ability isolated in China. Whole-genome sequencing on 2,212 Chinese isolates and 1,739 publicly available genomes reveals the population structure and evolutionary history of the rough colony variants. Characterized by macro, red, dry, and rough (mrdar) colonies, these variants demonstrate enhanced biofilm formation at 28 °C and 37 °C compared to typical rdar colonies. The mrdar variants exhibit extensive multidrug resistance, with significantly higher resistance to at least five classes of antimicrobial agents compared to non-mrdar variants. This resistance is primarily conferred by an IncHI2 plasmid harboring 19 antimicrobial resistance genes. Phylogenomic analysis divides the global collections into six lineages. The majority of mrdar variants belong to sublineage L6.5, which originated from Chinese smooth colony strains and possibly emerged circa 1977. Among the mrdar variants, upregulation of the csgDEFG operons is observed, probably due to a distinct point mutation (−44G > T) in the csgD gene promoter. Pangenome and genome-wide association analyses identify 87 specific accessory genes and 72 distinct single nucleotide polymorphisms associated with the mrdar morphotype.

Radial basis point interpolation for strain field calculation in digital image correlation

In order to extract smooth and accurate strain fields from the noisy displacement fields obtained by digital image correlation (DIC), a point interpolation meshless (PIM) method with a radial basis function (RBF) is introduced for full-field strain calculation, which overcomes the problems of slow calculation speed and unstable matrix inverse calculation of the element-free Galerkin method (EFG). The radial basis point interpolation method (RPIM) with three different radial basis functions and the moving least squares (MLS) and pointwise least squares (PLS) methods are compared by analyzing and validating the strain fields with high-strain gradients in simulation experiments. The results indicate that the RPIM is nearly 80% more computationally efficient than the MLS method when a larger support domain is used, and the efficiency of the RPIM is nearly 26% higher than that of the MLS method when a smaller support domain is used; the strain calculation accuracy is slightly lower than that of the MLS method by 0.3–0.5%, but the stability of the calculation is significantly improved. In contrast with the PLS method, which is easily affected by the noise and the size of the strain calculation window, the RPIM is insensitive to the displacement noise and the size of the support domain and can obtain a similar calculation accuracy. The RPIM with multiquadric (MQ) radial basis functions performs well in balancing the computational accuracy and efficiency and is insensitive to shape parameters. The application cases show that the method can effectively compute the strain field at the crack tip, validating its applicability to the study of the plastic region at the crack tip. In conclusion, the proposed RPIM-based method provides an accurate, practical, and robust approach for full-field strain measurements.

Multi-material topology optimization with a direct meshless local Petrov–Galerkin method applied to a two-dimensional boundary value problem

ABSTRACT This work presents a novel approach for the solution of multi-material topology optimization problems of elastic structures coupling the Direct Meshless Local Petrov–Galerkin (DMLPG) method with the Gradual Bi-direction Evolutionary Structural Optimization (gBESO) method. In recent years, meshless methods have been used in several engineering fields, showing some advantages compared to mesh-based methods. DMLPG is a truly meshless method, which achieves results with good accuracy and computational efficiency, avoiding the numerical integration of complicated shape functions by adopting low-degree polynomials. In the proposed algorithm, DMLPG is used to obtain smooth nodal displacements, strains and stresses, and gBESO updates the structural geometry based on design sensitivity values of the compliance while it gradually increases the Young's modulus of the material. The optimization problem minimizes the compliance objective function subject to volume constraints. Numerical examples demonstrate the applicability and validity of the technique.

Low-order finite element method for the static and vibration analysis of an antenna pedestal

In this work, an effective algorithm is developed for the static and vibration analysis of antenna pedestal, which can well balance the modelling efficiency and computational accuracy. At first, the antenna pedestal is discretized into a set of low-order tetrahedral elements that can be automatically generated. Then, the face-based smoothing domains are constructed based on the faces of the low-order tetrahedral mesh. With the help of strain smoothing technique, the system stiffness matrices are calculated in these smoothing domains and finally, the static and vibration responses of the antenna pedestal are obtained based on the related system equations. There is no need to add any penalty parameters or additional degrees of freedom. Numerical results demonstrate that the present algorithm can achieve much better accuracy and higher convergence rate than traditional FEM, which can be regarded as an efficient tool for the structural analysis of antenna pedestal.

A novel stabilized nodal integration formulation using particle finite element method for incompressible flow analysis

AbstractIn simulations using the particle finite element method (PFEM) with node‐based strain smoothing technique (NS‐PFEM) to simulate the incompressible flow, spatial and temporal instabilities have been identified as crucial problems. Accordingly, this study presents a stabilized NS‐PFEM‐FIC formulation to simulate an incompressible fluid with free‐surface flow. In the proposed approach, (1) stabilization is achieved by implementing the gradient strain field in place of the constant strain field over the smoothing domains, handling spatial and temporal instabilities in direct nodal integration; (2) the finite increment calculus (FIC) stabilization terms are added using nodal integration, and a three‐step fractional step method is adopted to update pressures and velocities; and (3) a novel slip boundary with the predictor–corrector algorithm is developed to deal with the interaction between the free‐surface flow with rigid walls, avoiding the pressure concentration induced by standard no‐slip condition. The proposed stabilized NS‐PFEM‐FIC is validated via several classical numerical cases (hydrostatic test, water jet impinging, water dam break, and water dam break on a rigid obstacle). Comparisons of all simulations to the experimental results and other numerical solutions reveal good agreement, demonstrating the strong ability of the proposed stabilized NS‐PFEM‐FIC to solve incompressible free‐surface flow with high accuracy and promising application prospects.

Slope Stability Analysis Based on the Explicit Smoothed Particle Finite Element Method

A landslide is a common natural disaster that causes environmental damage, casualties and economic losses, which seriously affects the sustainable development of society. In geomechanics, it is one of the largest deformation problems. Herein, the GPU-accelerated explicit smoothed particle finite element method (eSPFEM) for large deformation analysis in geomechanics was developed on the CUDA platform based on high-performance computing using a self-designed eSPFEM program code. The eSPFEM combines the strain smoothing nodal integration techniques found in the particle finite element method (PFEM) framework, which allows for the use of low-order triangular elements without volume locking and avoids frequent information transfer and mapping errors between Gaussian points and particles in PFEM. A numerical simulation of slope instability using the eSPFEM and based on a strength reduction technique was conducted using various examples, including a cohesive homogeneous slope, a non-cohesive homogeneous slope, a non-homogeneous slope and a slope with a thin soft band. The calculation results show that the eSPFEM can be applied to slope stability analysis under different working conditions, simulating the entire process of slope instability initiation, sliding and reaccumulation, and obtaining reliable FOS values. A numerical simulation was conducted to analyse a landslide that occurred in the Zhangjiazhuang tunnel on the Lanzhou–Xinjiang high-speed railway line on 18 January 2016. A natural unsaturated soil slope, a soil slope with a high moisture content and a soil slope with a high moisture content subjected to an earthquake were analysed. The findings of this study are in good agreement with the actual slope failure conditions. The primary triggers identified for the landslide were heavy rainfall and earthquakes. The verification results indicate that the eSPFEM can effectively simulate an actual landslide case, showcasing high accuracy and applicability in simulating the large deformation behaviour of landslides.

A stable implicit nodal integration-based particle finite element method (N-PFEM) for modelling saturated soil dynamics

In this study, we present a novel nodal integration-based particle finite element method (N-PFEM) designed for the dynamic analysis of saturated soils. Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner (HR) variational principle, creating an implicit PFEM formulation. To mitigate the volumetric locking issue in low-order elements, we employ a node-based strain smoothing technique. By discretising field variables at the centre of smoothing cells, we achieve nodal integration over cells, eliminating the need for sophisticated mapping operations after re-meshing in the PFEM. We express the discretised governing equations as a min-max optimisation problem, which is further reformulated as a standard second-order cone programming (SOCP) problem. Stresses, pore water pressure, and displacements are simultaneously determined using the advanced primal-dual interior point method. Consequently, our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation. Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy, obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches. This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

An accurate and robust strain field smoothing method based on polynomial fitting and anisotropic diffusion in digital image correlation

Digital Image Correlation (DIC) is a popular optical measurement method. However, the accuracy of strain values obtained by DIC may be adversely affected by random noise. This paper proposes an Anisotropic Nonlinear Diffusion Filtering (ADF) method, combining Pointwise Least Square (PLS) with the Perona-Malik (PM) model, extracts accurate and smooth strain fields from noisy displacement fields. The strain field is extracted point by point using PLS and then smoothed by the PM model. The ADF method was tested on simulated and real speckle deformation experiments. The results indicate that the ADF method significantly reduces the negative impact of inappropriate calculation parameters, such as the strain window size, compared to the PLS method. Furthermore, the PM model was found to be better suited for smoothing strain fields compared to other image denoising models. The ADF method demonstrates excellent noise suppression performance and accurate strain identification in both simulated and real experiments.

Convolution finite element based digital image correlation for displacement and strain measurements

This work presents a novel global digital image correlation (DIC) method, based on a newly developed convolution finite element (C-FE) approximation. The convolution approximation can rely on the mesh of linear finite elements and enables arbitrarily high order approximations without adding more degrees of freedom. Therefore, the C-FE based DIC can be more accurate than the usual FE based DIC by providing highly smooth and accurate displacement and strain results with the same element size. The detailed formulation and implementation of the method have been discussed in this work. The controlling parameters in the method include polynomial order, patch size, and dilation. A general choice of the parameters and their potential adaptivity have been discussed. The proposed DIC method has been tested by several representative examples, including the DIC challenge 2.0 benchmark problems, with comparison to the usual FE based DIC. C-FE outperformed FE in all the DIC results for the tested examples. This work demonstrates the potential of C-FE and opens a new avenue to enable highly smooth, accurate, and robust DIC analysis for full-field displacement and strain measurements.

Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods

The need to design structures and structural elements that are more efficient in terms of performance is a key aspect of engineering. For a given material to be used at its maximum capacity, considering non-linear characteristics is mandatory. The non-linear regime is a subject of extreme interest for this reason and is an area with intense research activity. In this work, advanced discretization techniques (i.e., meshless methods) are applied in the elastoplastic analysis of 2D and 3D structural elements. The literature shows that meshless methods are capable of producing more accurate and smoother strain and stress fields, which are the variable fields required in the non-linear models describing elastoplasticity. Thus, in this study, the Radial Point Interpolation Method (RPIM) and the Natural Neighbor Radial Point Interpolation Method (NNRPIM) are combined with a non-linear iterative algorithm, fully developed by the authors, with the objective of analyzing for the first time the elastoplastic behavior of a two-bay asymmetric frame and bowstring bridge considering 2D and 3D analysis. The accuracy and robustness of the RPIM and the NNRPIM are shown in the end, comparing the obtained results with FEM solutions and the available literature.

Effects of polydimethylsiloxane membrane surface treatments on human uterine smooth muscle cell strain response

In the United States, 1 in 10 infants are born preterm. The majority of neonatal deaths and nearly a third of infant deaths are linked to preterm birth. Preterm birth is initiated when the quiescent state of the uterus ends prematurely, leading to contractions and parturition beginning as early as 32 weeks, though the origins are not well understood. To enable research and discovery of therapeutics with potential to better address preterm birth, the capability to study isolated cell processes of pregnant uterine tissue in vitro is needed. Our development of an in vitro model of the myometrium utilizing human uterine smooth muscle cells (uSMCs) responsible for contractions provides a methodology to examine cellular mechanisms of late-stage pregnancy potentially involved in preterm birth. We discuss culture of uSMCs on a flexible polydimethylsiloxane (PDMS) substrate functionalized with cationic poly-l-lysine (PLL), followed by extracellular matrix (ECM) protein coating. Previous work exploring uSMC behavior on PDMS substrates have utilized collagen-I coatings, however, we demonstrated the first exploration of human uSMC response to strain on fibronectin-coated flexible membranes, importantly reflecting the significant increase of fibronectin content found in the myometrial ECM during late-stage pregnancy. Using the model we developed, we conducted proof-of-concept studies to investigate the impact of substrate strain on uSMC cell morphology and gene expression. It was found that PLL and varied ECM protein coatings (collagen I, collagen III, and fibronectin) altered cell nuclei morphology and density on PDMS substrates. Additionally, varied strain rates applied to uSMC substrates significantly impacted uSMC gene expression of IL-6, a cytokine associated with instances of preterm labor. These results suggest that both surface and mechanical properties of in vitro systems impact primary human uSMC phenotype and offer uSMC culture methodologies that can be utilized to further the understanding of cellular pathways involved in the uterus under mechanical load.

Evaluation of distributed fibre optic sensors in structural concrete

In civil engineering, the analysis and characterisation of structural phenomena is key for the deviation of mechanical models and strongly depends on experimental studies. For this reason, the development of new measurement techniques plays a crucial role in research to allow capturing the mechanical behaviour of structures. In this context, distributed fibre optic sensing (DFOS) gained attention during the last years. Advantageous properties such as minimal invasiveness and quasi-continuous strain measurement enable new possibilities in structural monitoring. DFOS is able to sense minimal strain variations, yet this often results in recording unwanted anomalies. Thus, to generate the best results from raw measurement data, it is crucial to use a robust and reliable post-processing procedure. In order to meet this requirement, the software solution FOS Evaluator was developed using the Python programming language, targeting the most important aspects that may occur when using DFOS. Those aspects comprise a high data volume, disturbances and anomalies in the measured data and the necessity to carry out calculations with acquired data sets. In this paper, functionality and background of FOS Evaluator are presented and discussed. Furthermore, several methods for post-processing and evaluating DFOS measurements are presented, consisting of multiple functions for reducing, cleaning, or smoothing strain data, and calculating various mechanical properties from filtered measurements. Finally, the functional scope of FOS Evaluator is illustrated by various application examples in structural concrete.

The small non-coding RNA B11 regulates multiple facets of Mycobacterium abscessus virulence.

Mycobacterium abscessus causes severe disease in patients with cystic fibrosis. Little is known in M. abscessus about the roles of small regulatory RNAs (sRNA) in gene regulation. We show that the sRNA B11 controls gene expression and virulence-associated phenotypes in this pathogen. B11 deletion from the smooth strain ATCC_19977 produced a rough strain, increased pro-inflammatory signaling and virulence in multiple infection models, and increased resistance to antibiotics. Examination of clinical isolate cohorts identified isolates with B11 mutations or reduced expression. We used RNAseq and proteomics to investigate the effects of B11 on gene expression and test the impact of mutations found in clinical isolates. Over 200 genes were differentially expressed in the deletion mutant. Strains with the clinical B11 mutations showed expression trends similar to the deletion mutant, suggesting partial loss of function. Among genes upregulated in the B11 mutant, there was a strong enrichment for genes with B11-complementary sequences in their predicted ribosome binding sites (RBS), consistent with B11 functioning as a negative regulator that represses translation via base-pairing to RBSs. Comparing the proteomes similarly revealed that upregulated proteins were strongly enriched for B11-complementary sequences. Intriguingly, genes upregulated in the absence of B11 included components of the ESX-4 secretion system, critical for M. abscessus virulence. Many of these genes had B11-complementary sequences at their RBSs, which we show is sufficient to mediate repression by B11 through direct binding. Altogether, our data show that B11 acts as a direct negative regulator and mediates (likely indirect) positive regulation with pleiotropic effects on gene expression and clinically important phenotypes in M. abscessus. The presence of hypomorphic B11 mutations in clinical strains is consistent with the idea that lower B11 activity may be advantageous for M. abscessus in some clinical contexts. This is the first report on an sRNA role in M. abscessus.

Adaptive quadtree polygonal based edge-based smoothed finite element method for quasi-incompressible hyperelastic solids

This paper discusses an adaptive framework based on the edge-based strain smoothing approach with polygonal meshes for largely deformable quasi-incompressible hyperelasticity. The proposed approach employs the quadtree decomposition for spatial discretization and the strain smoothing technique to compute the bilinear/linear form. The local refinement is based on the stress distribution and the element that has hanging nodes due to adaptive local refinement are treated as polygonal element within the strain smoothing framework. The accuracy and the robustness of the proposed framework are numerically studied with a few examples. When compared to uniform refinement, it is seen that the proposed framework yields comparable results with fewer degrees of freedom.

A centroid-enriched strain-smoothed radial point interpolation method for nearly incompressible elastoplastic problems in solid mechanics

While edge-based (2D) and face-based (3D) smoothed radial point interpolation methods (ES-RPIM and FS-RPIM) have demonstrated excellent performance in solid mechanics, the volumetric locking problem limits their applications in nearly incompressible problems. This paper develops a centroid-enriched edge-based/face-based smoothed RPIM (CE-ES-RPIM and CE-FS-RPIM) to address the volumetric locking problem in nearly incompressible problems. The proposed method comprises the following elements: (1) a centroid-enriched scheme is creatively used and implemented into the traditional ES-RPIM and FS-RPIM framework, in which the centroids of triangular mesh are added as spatial discretization nodes; (2) the strain-smoothing technique is performed on triangle (2D) or tetrahedron (3D) mesh to maintain the linear exactness of the radial point interpolation method; and (3) the combination of the centroid-enriched scheme and strain smoothing technique makes the ratio of the number of displacement equations to the number of incompressible constraints reach the optimal value of 2, effectively eliminating volumetric locking. The accuracy, effectiveness and efficiency of the proposed method are validated on several 2D and 3D nearly incompressible benchmark examples. All results demonstrate that the proposed method can provide accurate numerical results under an incompressible limitation and provide better performance than conventional quadratic element methods.