Natural Neighbour Radial Point Interpolation Method Research Articles

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.

Total knee arthroplasty coronal alignment and tibial base stress-a new numerical evaluation.

Total knee arthroplasty (TKA) is one of the most frequently performed orthopedic procedures. The correct positioning and alignment of the components significantly affects prosthesis survival. Considering the current controversy regarding the target of postoperative alignment of TKA, this study evaluated the tension at tibial component interface using two numerical methods. The stress of the prosthesis/bone interface of the proximal tibial component was evaluated using two numerical methods: the finite element method (FEM) and the new meshless method: natural neighbor radial point interpolation method (NNRPIM). The construction of the model was based on Zimmers NexGen LPS-Flex Mobile® prosthesis and simulated the forces by using a free-body diagram. Tibiofemoral mechanical axis (TFMA) for which a higher number of nodes are under optimal mechanical tension is between 1° valgus 2° varus. For values outside the interval, there are regions under the tibial plate at risk of bone absorption. At the extremities of the tibial plate of the prosthesis, both medial and lateral, independent of the alignment, are under a low stress. In all nodes evaluated for all TFMA, the values of the effective stresses were higher in the NNRPIM when compared with the FEM. Through this study, we can corroborate that the optimal postoperative alignment is within the values that are currently considered of 0 ± 3° varus. It was verified that the meshless methods obtain smoother and more conservative results, which may make them safer when transposed to the clinical practice.

Topology optimization using a natural neighbour meshless method combined with a bi-directional evolutionary algorithm

Due to recent developments in the additive manufacturing industry, topology optimization is nowadays a powerful computational tool that allows to design feasible lightweight components. Although the Finite Element Method (FEM) is the most applied discretization technique, meshless methods are currently established as accurate numerical methods with relevant advantages in several engineering fields. Nonetheless, the state-of-the-art of meshless methods in topology optimization is still scarce. This work develops the combination of a bi-direction structural optimization (BESO) algorithm with the Natural Neighbour Radial Point Interpolation Method (NNRPIM), a meshless method combining the natural neighbours geometric concept with the RPI shape functions. First, several benchmark examples are solved to evaluate the algorithm capability under several algorithm parameters. The proposed methodology is then implemented to design new automotive lightweight components. The results from the numerical applications demonstrate that the NNRPIM is a solid technique to be incorporated in optimization algorithms. Additionally, innovative automotive industry designs for additive manufacturing can be obtained using the presented approach.

A bio-inspired remodelling algorithm combined with a natural neighbour meshless method to obtain optimized functionally graded materials

Recent developments suggest the use of triply periodic minimal surfaces (such as the gyroid) as a possibility for bone tissue scaffold. Moreover, through functional gradients of cellular structures, the mechanical properties can be edited and enhanced to achieve the most efficient results. One of the main concerns when designing bone scaffold is avoiding stress shielding, which occurs when the Young’s modulus of the implant is higher than the Young’s modulus of the bone it is replacing. If so, bone decay occurs in the surrounding tissue. While the literature possesses some approaches exploring functional gradients of material density, there are no solutions based on bone tissue phenomenological laws. Thus, the gyroid infill obtained with PLA (E=3145 MPa) was characterized with mechanical tests, namely tensile and compression, and the obtained model was implemented in a bone remodelling algorithm. Using the natural neighbour radial point interpolation method (NNRPIM) it was found that similar bone density distributions were obtained for the gyroid infill and for bone tissue when subject to the same boundary conditions. Finally, the gyroid mechanical behaviour was extrapolated to other materials and it was concluded that similar properties can be obtained for bone tissue and titanium alloy (E=110GPa) scaffold.

Modelling adhesively-bonded T-joints by a meshless method

Bonding of non-parallel substrates has many applications in the transport industry. Adhesively-bonded T-joints are employed for that purpose. However, geometrical dimensions, substrates’ shape and material choices are broad. The effect that varying upper substrate thickness has on the joint strength (P max ) was investigated in this work through numerical analyses. The numerical analysis was performed using a meshless method, the natural neighbour radial point interpolation method (NNRPIM), which has been proven accurate and robust in another adhesive joint configuration. Materials were considered elastic-plastic. A yield criterion developed for rubber-like materials, the Exponent Drucker-Prager, was used for the adhesive layer, while the metallic substrates were analysed with the von Mises yield criterion. P max was determined numerically using a strain-based continuum mechanics failure criterion. Normalised peel and shear stress distributions along the bond-line are presented. Effective strains, both elastic and plastic, were also obtained. The estimated P max was compared with experimental data; a good agreement was found. The stress distribution along the bond line becomes asymmetric in joints with unbalanced substrate thicknesses. At P max , from 10 to 25% of the bond-line has entered into plastic regime. The results indicate that the proposed methodology is suitable to analyse adhesively-bonded joints under different load solicitations.

The natural neighbor radial point interpolation method in the Elasto-Static analysis of Honeycomb-Shaped foams

Cellular solid materials are progressively becoming more predominant in lightweight structural applications as more technologies realize these materials can be improved in terms of performance, quality control, repeatability and production costs, when allied with fast developing manufacturing technologies such as Additive Manufacturing. In parallel, the rapid advances in computational power and the use of new numerical methods, such as Meshless Methods, in addition to the Finite Element Method (FEM) are highly beneficial and allow for more accurate studies of a wide range of topologies associated with the architecture of cellular solid materials. Since these materials are commonly used as the cores of sandwich panels, in this work, two different topologies were designed — conventional honeycombs and re-entrant honeycombs — for 7 different values of relative density, and tested on the linear-elastic domain, in both in-plane directions, using the Natural Neighbor Radial Point Interpolation Method (NNRPIM), a newly developed meshless method, and the Finite Element Method (FEM) for comparison purposes.

Meshless and hyper-elastic implementation to analyse flexible adhesives

Structural adhesives and the joints involving them are often analysed using elastic and elastic-plastic material models. Nowadays, flexible adhesives are being used for structural applications. The material models used for structural adhesives are not always suitable for flexible adhesives. Hyper-elastic material models had been proven more suitable to model flexible adhesives. The Finite Element Method (FEM) is a well-accepted technique to model and analyse adhesive joints. Commercial FEM packages already have hyper-elastic capabilities, but not many applications to adhesive joints had been studied. Recently, Meshless Methods (MM) have also been brought into this field. Although the nature of MM made them attractive for the analysis of flexible adhesives and the joints bonded with them, no previous analyses had been reported. Several hyper-elastic models are available. One of them, the Ogden model, can represent several models through converting their parameters. Therefore, the Ogden model implementation is more versatile.In this work, the Ogden hyper-elastic model was implemented into a MM, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). The implementation was validated using literature data. The proposed methodology could be the basis for further application to adhesive joints bonded with flexible adhesives.

The numerical analysis of symmetric cross-ply laminates using the natural neighbour radial point interpolation method and high-order shear deformation theories

Composite structures are commonly analysed using the Finite Element Method (FEM). However, new accurate and efficient discrete numerical techniques have appeared recently – the meshless methods. Thus, this work uses a meshless method – the Natural Neighbour Radial Point Interpolation Method (NNRPIM) – to perform an elasto-static analysis of composite laminated plates. Meshless methods only require an unstructured nodal distribution to discretize the problem domain. In order to numerically integrate the integro-differential equation from the Galerkin weak formulation, a background integration mesh is constructed using the Voronoï diagram. Then, the nodal connectivity is enforced using the ‘influence-cell’ concept and the shape functions are obtained. In this work, laminated composite plates are analysed using distinct equivalent single layer theories, considering different transverse high-order shear deformation laws. Thus, several third-order, exponential and trigonometric transverse shear deformation theories are combined with the NNRPIM to analyse the structural response of composite laminated plates. In the end, composite laminated plates are numerically analysed and the meshless solutions are compared with the analytical solution available in the literature. Therefore, this works contributes with new solutions for classic composite symmetric cross-ply laminated plates and provides a comparative study on the accuracy of some high-order shear deformation theories (HSDTs).

Analysis of antisymmetric cross-ply laminates using high-order shear deformation theories: a meshless approach

For many years finite element method (FEM) was the chosen numerical method for the analysis of composite structures. However, in the last 20 years, the scientific community has witnessed the birth and development of several meshless methods, which are more flexible and equally accurate numerical methods. The meshless method used in this work is the natural neighbour radial point interpolation method (NNRPIM). In order to discretize the problem domain, the NNRPIM only requires an unstructured nodal distribution. Then, using the Voronoi mathematical concept, it enforces the nodal connectivity and constructs the background integration mesh. The NNRPIM shape functions are constructed using the radial point interpolation technique. In this work, the displacement field of composite laminated plates is defined by high-order shear deformation theories. In the end, several antisymmetric cross-ply laminates were analysed and the NNRPIM solutions were compared with the literature. The obtained results show the efficiency and accuracy of the NNRPIM formulation.

Material non-linearity in the numerical analysis of SLJ bonded with ductile adhesives: A meshless approach

Modern transportation, medical, and leisure equipment are often manufactured with non-metallic and composite materials bonded with adhesives. Numerical analyses of these joints are commonly performed using the Finite Element Method (FEM). In adhesive joints, the bond line is small compared to the substrates requiring highly refined meshes to perform the analyses. Also, adhesives can reach higher strains compared to the substrates, and so causing mesh distortion and compromising the solution. Alternatively, meshless methods (MM) can be employed. Single-lap joints (SLJ), during loading, present tensile and compressive stresses along the adhesive layer. Commonly, in elastic-plastic analyses, equal adhesive's strength in tension and compression is considered (using von Mises (vM) yield criterion), but it has been reported that adhesives are stronger in compression. Thus, a yielding criterion such as the Exponent Drucker-Prager (EDP) can be used; however, no MM implementations are available in the literature. This work seeks to implement the EDP into a MM, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). Then, using the NNRPIM elastic-plastic parametric analyses of SLJ with aluminium substrates were performed considering four overlap lengths (LO) and two different ductile adhesives. All the cases were also evaluated using MM and the vM yield criterion, as a benchmark. Stress and strain distributions along the bond-line were obtained; afterwards, joint strength (Pmax) was determined by using continuum mechanics failure criteria, and then, evaluated and compared. Then, the results were compared against experimental data. The cases solved using the EDP predicted Pmax closer to the experimental data; the joints with intermediate overlap lengths LO=25.0 mm presented the closest values. In conclusion, the results' correlation with experimental data indicates the EDP is closer than vM to the actual adhesive behaviour in the joints. Also, the EDP was successfully implemented. Overall, the NNRPIM is a strong and accurate alternative for elastic-plastic analyses of adhesive joints.

A meshless study of antisymmetric angle-ply laminates using high-order shear deformation theories

Composite laminated plates have found several applications in engineering field. For instance, aerospace and automotive industry use these structures in several components of their products due to their exceptional specific properties along the direction of the fibres. There are several two-dimensional plate models that aim to more accurately and more efficiently predict the kinematics of those structures. Unlike the Classical Plate Theory (CLPT), which can only be applied to thin laminates, the First-Order Shear Deformation Theory (FSDT), developed by Reissner and Mindlin, already accounts shear effects and, due to that, it can describe in a satisfactory way the kinematic of a generic thick laminated plate. Nevertheless, the FSDT considers linear shape functions to describe the in-plane displacements through the plate thickness, which results in constant shear strains. Thus, the traction boundary conditions on the top and bottom surface of plate are violated. Additionally, the FSDT needs shear correction factors, which are not easily obtained for a generic problem. Equivalent single layer (ESL) theories following high-order shear deformation theories (HSDTs) have been proposed and more efficiently applied to composite laminated plates. These theories allow to represent the nonlinear variation of transverse shear stresses through thickness of plate because they possess nonlinear shape functions interpolating the in-plane displacements. Thus, the kinematics of these structural elements can be better described and more accurate stress fields can be obtained for problems that cannot be handled with the FSDT.These structures have been analysed for several years using the Finite Element Method (FEM), which is the most popular numerical tool in structural analysis. However, several other advanced numerical techniques, such as meshless methods, can also handle this kind of problems and, in some cases, in a more efficient way. This work makes use of a recently developed meshless method – the Natural Neighbour Radial Point Interpolation Method (NNRPIM) (Belinha, 2014) – to study the bending of antisymmetric angle-ply composite laminates using distinct HSDTs.

Analyzing single-lap joints bonded with a brittle adhesive by an elastic meshless method

Adhesive bonding is a widespread joining technique in the industry, namely aeronautical and automotive. However, the associated products often have complex and non-standard shapes. Numerical modeling, such as that based on the finite element method (FEM), is commonly applied for bonded joint design, although there are still limitations linked to the method. Meshless methods (MM) are a viable option to overcome some limitations of the FEM. However, these methods are still in development.In this work, a robust meshless method, the Natural Neighbor Radial Point Interpolation Method (NNRPIM), was applied to the analysis of adhesive joints. First, experimental data of single-lap joints (SLJ) corresponding to four overlap lengths (LO) and a brittle adhesive were measured as a benchmark. Then, all the joint geometries were simulated utilizing the FEM and NNRPIM methodologies. The NNRPIM proved to be a good alternative to the FEM, providing similar strength prediction with differences less than 3% between them. Moreover, the stress distribution curves were compared, and differences of 6% in the peak shear stresses were found. In conclusion, the NNRPIM provides accurate results and could be utilized for further study of adhesively bonded joints.

The numerical simulation of crack propagation using radial point interpolation meshless methods

In this work, a new crack propagation prediction algorithm is combined with two distinct meshless methods – the Radial Point Interpolation Method (RPIM) and the Natural Neighbor Radial Point Interpolation Method (NNRPIM). Both these advanced discretization numerical techniques construct the shape functions using the radial point interpolating technique, which allows to build shape functions possessing the delta Kronecker property. The RPIM requires a background integration grid to numerically integrate the partial differential equations ruling the physical phenomenon. Alternatively, the NNRPIM constructs the background integration points using only the information of the spatial position of the nodes.This work aims to present a new methodology to predict the crack propagation and, simultaneously, to compare the performance of both meshless techniques (for similar conditions). The developed algorithm advances the crack tip iteratively and uses the maximum tangential stress criterion to calculate the propagation direction. In the end, three benchmark tests were analyzed using the proposed algorithm. Accurate crack paths were obtained when compared to the solutions described in the literature.

A new biological bone remodeling in silico model combined with advanced discretization methods.

Bone remodeling remains a highly researched topic investigated by many strands of science. The main purpose of this work is formulating a new computational framework for biological simulation, extending the version of the bone remodeling model previously proposed by Komarova. Thus, considering only the biological aspect of the remodeling process, the action of osteoclasts and osteoblasts is taken into account as well as its impact on bone mass. It is conducted a spatiotemporal analysis of a remodeling cycle obtaining a dynamic behavior of bone cells very similar to the biological process already described in the literature. The numerical example used is based on bone images obtained with scanning electron microscopy. During simulation, it is possible to observe the variation of bone's architecture through isomaps. These maps are obtained through the combination of biological bone remodeling models with three distinct numerical techniques-finite element method (FEM), radial point interpolation method (RPIM), and natural neighbor radial point interpolation method (NNRPIM). A study combining these numerical techniques allows to compare their performance. Ultimately, this work supports the inclusion of meshless methods due to their smoother results and its easiness to be combined with medical images from CT scans and MRI.

A new numerical approach to mechanically analyse biological structures

In this work, an advanced discretization meshless technique is used to study the structural response of a human brain due to an impact load. The 2D and 3D brain geometrical models, and surrounding structures, were obtained through the processing of medical images, allowing to achieve a realistic geometry for the virtual model and to define the distribution of the mechanical properties accordingly with the medical images colour scale. Additionally, a set of essential and natural boundary conditions were assumed in order to reproduce a sudden impact force applied to the cranium. Then, a structural numerical analysis was performed using the Natural Neighbour Radial Point Interpolation Method (NNRPIM). The obtained results were compared with the finite element method (FEM) and a solution available in the literature. This work shows that the NNRPIM is a robust and accurate numerical technique, capable to produce results very close to other numerical approaches. In addition, the variable fields obtained with the meshless method are much smoother than the FEM corresponding solution.

A multiscale homogenization procedure combining the fabric tensor with a natural neighbour meshless method

Bone is a material that can be classified as a hierarchical structure, where the different structural levels can be identified from the microscale to macroscale. Multiscale models enable to model the material using homogenization techniques. In this work a innovative homogenization technique for trabecular bone tissue is proposed, which uses the fabric tensor concept and a bone phenomenological material law, linking the apparent density with the trabecular bone mechanical proprieties. The proposed methodology efficiently homogenize the trabecular bone highly heterogeneous medium, allowing to define its homogenized microscale mechanical properties and to reduce the analysis computational cost (when compared with classical homogenization techniques). The homogenization technique is combined with the natural neighbour radial point interpolation method (NNRPIM). The NNRPIM uses the natural neighbour mathematical concept to enforce the nodal connectivity and build the background integration mesh, required to numerically integrate the integro-differential elasticity equations. Furthermore, the NNRPIM uses the radial point interpolators technique to construct its interpolating shape functions. In order to verify the efficiency of the technique several examples are solved using a confined square patch of trabecular bone under compression. In the end, the results obtained with a classic homogenization technique and the proposed methodology are compared.

The free vibrations analysis of the cupula in the inner ear using a natural neighbor meshless method

The cupula is a part of the inner ear semi-circular canals that plays an important role in the maintenance of the human balance. In order to understand the dynamic response of the cupula, first it is necessary to obtain its vibration frequencies. A two-dimensional and three-dimensional geometrical model of the cupula was built. Then, a free vibration analysis was performed using two distinct numerical techniques, the finite element method (FEM) and the natural neighbor radial point interpolation method (NNRPIM). Besides the fundamental analysis, other scenarios were studied, aiming to analyze the environment of the cupula (in healthy and pathologic scenarios). The results obtained with the geometrical models show that NNRPIM is capable to deliver results very close with the FEM. Additionally, the NNRPIM formulation possesses a high convergence and acceptable computational costs.This work presents for the first time a computational study on the free vibration analysis of the cupula and shows an alternative numerical technique to calculate with precision the natural frequency of the cupula. The outcomes of this work will allow the development of alternative therapies for cupulolithiasis, which causes severe dizziness.

Mechanical bone remodelling: Comparative study of distinct numerical approaches

Bone is a complex living structure. Induced by its biomechanical environment, bone constantly adapts its internal and external structure, altering its mass and morphology. In this work, three distinct discretization techniques are combined with a mechanologic bone tissue remodelling algorithm, which describes the remodelling process by minimizing the strain energy density (SED) field. During an iterative remodelling process, bone tissue's mechanical properties are recurrently correlated with bone apparent density through a phenomenological law. The purpose of this study is to compare bone tissue remodelling solutions obtained with the Finite Element Method (FEM), the Radial Point Interpolation Method (RPIM) and the Natural Neighbour RPIM (NNRPIM). Thus, two numerical examples were analysed, namely a benchmark bone patch and a femur. On both examples, regardless the discretization technique, it was possible to accurately reproduce the trabecular bone architecture. Additionally, a comparison between each numerical method was performed. The model shown high adaptability, since, for every method, successful performances were obtained. Nevertheless, meshless solutions produced smoother and more accurate results, closer to real X-ray images.

Homogenization technique for heterogeneous composite materials using meshless methods

Due to its random fibre distribution across the cross-section and their anisotropic and heterogeneous characteristic, the prediction of the mechanical behaviour of fibre composite materials is complex. Multi-scale approaches have been proposed in the literature to more accurately predict their mechanical properties using computational homogenization procedures.This work is based on existing multi-scale numerical transition techniques suitable for simulating heterogeneous materials and makes use of two meshless methods—the Radial Point Interpolation Method (RPIM) and the Natural Neighbour Radial Point Interpolation Method (NNRPIM)—and the Finite Element Method (FEM). Representative volume elements (RVEs) are modelled and discretized using the three numerical methods. Prescribed microscopic displacements are imposed on different RVEs whose boundaries are periodic and, from the obtained stress field, the average stresses are determined. Consequentially, the effective elastic properties of a heterogeneous material are obtained for different fibre volume fractions. In the end, the numerical solutions are compared with the solutions proposed in the literature and it is proved that the NNRPIM achieve more accurate solutions than the RPIM and the FEM.

Simulating fracture propagation in brittle materials using a meshless approach

In the present work, a meshless method is combined with a crack growth algorithm, considering a linear elastic model to simulate the propagation of cracks in brittle materials. The proposed methodology is automatic, which means that it does not require the user intervention. In this study, the meshless method used is the Natural Neighbour Radial Point Interpolation Method (NNRPIM), an efficient discrete numeric method. Being a truly meshless method, the NNRPIM only requires an unstructured nodal distribution to fully discretize the problem domain. With the coordinates of the nodes, the NNRPIM formulation is autonomously capable: to define the nodal connectivity; to build the background integration mesh; and to construct the shape functions. The crack propagation algorithm here proposed simulates the crack growth by displacing iteratively the crack tip. Updating the crack tip location requires a local remeshing, which does not represent a numeric difficulty for the NNRPIM. The updated position of the crack tip depends on the estimated stress field in the vicinity of the crack tip. Thus, in each iteration, the stress field in the vicinity of the crack tip is estimated, and then, the direction of the crack propagation is calculated using the maximum circumferential stress criterion.