L.M.J.S. Dinis

The natural neighbour radial point interpolation method: dynamic applications

Purpose – The purpose of this paper is to extend the natural neighbour radial point interpolation method (NNRPIM) to the dynamic analysis (free vibrations and forced vibrations) of two‐dimensional, three‐dimensional and bending plate problems.Design/methodology/approach – The NNRPIM shape‐function construction is briefly presented, as are the dynamic equations and the mode superposition method is used in the forced vibration analysis. Several benchmark examples of two‐dimensional and plate bending problems are solved and compared with the three‐dimensional NNRPIM formulation. The obtained results are compared with the available exact solutions and the finite element method (FEM) solutions.Findings – The developed NNRPIM approach is a good alternative to the FEM for the solution of dynamic problems, once the obtained results with the EFGM shows a high similarity with the obtained FEM results and for the majority of the studied examples the NNRPIM results are more close to the exact solution results.Researc...

Composite Laminated Plates: A 3D Natural Neighbor Radial Point Interpolation Method Approach

Based on the natural neighbor radial point interpolation method (NNRPIM), a 3D analysis of thick composite laminated plates is presented. The NNRPIM uses the natural neighbour concept in order to enforce nodal connectivity. Based on the Voronoï diagram small cells are created from the unstructured set of nodes discretizing the problem domain, the ‘influence-cells’, which are in fact influence domains entirely nodal dependent. The Delaunay triangles, the dual of the Voronoï cells, are used to create a node-dependent background mesh used in the numerical integration of the NNRPIM interpolation functions. The NNRPIM interpolation functions, used in the Galerkin weak form, are constructed in a process similar to that in the radial point interpolation method (RPIM) with some differences that modify the method performance. In the construction of the NNRPIM interpolation functions, no polynomial base is required and the used radial basis function (RBF) is the multiquadric RBF. The NNRPIM interpolation functions possess the delta Kronecker property, which simplifies the imposition of the natural and essential boundary conditions. In this work the 3D NNRPIM analysis is used to solve static and dynamic composite laminated plate problems. Thus, several benchmark examples are studied to demonstrate the effectiveness of the method.

An Unconstrained Third-Order Plate Theory Applied to Functionally Graded Plates Using a Meshless Method

An interpolator meshless method is used in the numerical implementation of an Unconstrained Third-Order Plate Theory applied to functionally graded plates. The meshless method enforces the nodal connectivity using the Natural Neighbor concept and uses the Radial Point Interpolators in order to construct the interpolation functions, which possess the delta Kronecker property. The meshless method uses the weak-form of Galerkin, which is integrated with a background integration mesh completely node dependent. Several static and dynamic functionally graded plate and sandwich plate examples are solved in order to prove the high accuracy and convergence rate of the proposed method.

A meshless microscale bone tissue trabecular remodelling analysis considering a new anisotropic bone tissue material law.

In this work, a novel anisotropic material law for the mechanical behaviour of the bone tissue is proposed. This new law, based on experimental data, permits to correlate the bone apparent density with the obtained level of stress. Combined with the proposed material law, a biomechanical model for predicting bone density distribution was developed, based on the assumption that the bone structure is a gradually self-optimising anisotropic biological material that maximises its own structural stiffness. The strain and the stress field required in the iterative remodelling process are obtained by means of an accurate meshless method, the Natural Neighbour Radial Point Interpolation Method (NNRPIM). Comparing with other numerical approaches, the inclusion of the NNRPIM presents numerous advantages such as the high accuracy and the smoother stress and strain field distribution. The natural neighbour concept permits to impose organically the nodal connectivity and facilitates the analysis of convex boundaries and extremely irregular meshes. The viability and efficiency of the model were tested on several trabecular benchmark patch examples. The results show that the pattern of the local bone apparent density distribution and the anisotropic bone behaviour predicted by the model for the microscale analysis are in good agreement with the expected structural architecture and bone apparent density distribution.

Elasto‐plastic analysis of plates by the element free Galerkin method

Purpose – The aim of this paper is to extend the Element Free Galerkin method (EFGM) in order to perform the elasto‐plastic analysis of isotropic plates.Design/methodology/approach – The EFGM shape‐function construction is briefly presented. The Newton‐Raphson method and the elasto‐plastic algorithm adapted to the EFGM, are described. Several plate bending non‐linear material problems are solved and the obtained solutions are compared with available finite element method (FEM) solutions.Findings – The paper finds that the developed EFGM approach is a good alternative to the FEM for the solution of non‐linear problems, once the obtained results with the EFGM show a high similarity with the obtained FEM results.Research limitations/implications – Comparing the FEM and the EFGM there are some drawbacks for the EFGM. The computational cost of the EFGM is higher, the imposition of the essential boundary conditions is more complex and there is a high sensitivity of the method in what concerns the choice of the ...

THE MANDIBLE REMODELING INDUCED BY DENTAL IMPLANTS: A MESHLESS APPROACH

This work aims to evaluate the prediction efficiency of a recently developed numerical approach in the mandible bone tissue remodeling process. The remodeling algorithm, seeking the minimization of the strain energy density, includes a phenomenological anisotropic material law capable of predicting the bone tissue mechanical properties based on the bone apparent density. The key factor of the proposed numerical approach is the inclusion of a flexible and efficient meshless method, which is used to obtain the strain energy density field. The inclusion of this advance discretization technique in the process is an asset since meshless methods produce smoother and more accurate strain energy density fields when compared with other numerical approaches. The bone tissue remodeling process of the molar region of the mandible, due to the inclusion of an implant system, is studied. The obtained results are in accordance with other numerical approach results available in the literature.

The Radial Natural Neighbours Interpolators Extended to ElastoplastiCity

Considering a small strain formulation a Radial Natural Neighbour Interpolator method is extended to the elastoplastic analysis. Resorting to the Voronoi tessellation the nodal connectivity is obtained. The Delaunay triangulation supplies the integration background mesh. The improved interpolation functions based on the Radial Point Interpolators are provided with the delta Kronecker property, easing the imposition of the essential and natural boundary conditions. The Newton-Raphson method is used for the solution of the nonlinear system of equations and an Hill yield surface is considered. Benchmark examples prove the high accuracy and convergence rate of the proposed method.

The Natural Neighbor Radial Point Interpolation Method Extended to the Crack Growth Simulation

In the present work, the Natural Neighbor Radial Point Interpolation Method (NNRPIM) is used to simulate the crack growth phenomenon in brittle materials. In order to discretize the problem domain, the NNRPIM only requires an unstructured nodal distribution. With the spatial information of the computational nodes, the NNRPIM is capable to automatically establish the nodal connectivity and to construct the interpolation functions. Additionally, using the natural neighbor geometrical concept, the NNRPIM is able to obtain, from the unstructured nodal distribution, the integration background mesh required to numerically integrate the integro-differential equations ruling the studied physical phenomenon. In this work, a crack opening path algorithm is adapted and combined with the NNRPIM. The developed algorithm is able to predict the crack growth by relocating iteratively the crack tip. The stress field in the vicinity of the crack tip is determined in each iteration and then, using the maximum circumferential stress criterion, the direction of the crack propagation is calculated. Here, the repositioning of the crack tip requires a local re-meshing. However, due to the flexibility of the natural neighbor concept, the local re-meshing do not represent a numerical difficulty. Additionally, in order to show the efficiency of the proposed approach, several demanding crack growth benchmark examples are solved.

The analysis of the bone remodelling around femoral stems

In this work the bone tissue remodelling analysis of the femur bone due to the insertion of a stem is analysed. The innovation of this numerical application lies on the use of an efficient advanced discretization meshless technique to obtain the variable fields required by the remodelling process.The remodelling algorithm used in this work allows to update locally and gradually the bone tissue density of the model. Based on the minimization of the strain energy density (SED), the remodelling algorithm uses a phenomenological bone tissue mathematical law, which permits to correlate the bone apparent density with its own mechanical properties. Comparing with other numerical approaches, the advanced discretization meshless technique used in this work allows to obtain smoother and more accurate SED fields, leading to more reliable remodelling predictions.The numerical remodelling results obtained for the femoral diaphysis after the insertion of the stem are in accordance with the clinical observation and with other numerical approaches results available in the literature.

The Dynamic Analysis of Thin Structures Using a Radial Interpolator Meshless Method

In this chapter an improvement of the Natural Neighbour Radial Point Interpolation Method (NNRPIM), a recently developed meshless method, is presented. A new approach of the NNRPIM is proposed, the NNRPIM 3D Shell-Like formulation, in order to analyse dynamically thin three-dimensional structures. The NNRPIM uses the Natural Neighbour concept to enforce the nodal connectivity and to construct the integration background mesh (totally node-dependent), which is used in the numerical integration of the NNRPIM interpolation functions. The essential and natural boundaries are imposed directly once the NNRPIM interpolation functions possess the delta Kronecker property. Several dynamic plate and shell problems are studied to demonstrate the effectiveness of the method.