Meshless Methods

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Meshless methods are numerical techniques used for solving partial differential equations without the need for a predefined mesh. They rely on a set of scattered points to represent the solution domain, allowing for greater flexibility in handling complex geometries and dynamic problems, while improving computational efficiency and accuracy.

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Key research themes

1. How have meshless numerical methods evolved to improve accuracy and computational efficiency for solving PDEs in complex geometries?

This research theme investigates the development and advancement of meshless methods as alternatives to traditional mesh-based techniques (like FEM and FDM) for discretizing and solving partial differential equations, focusing on performance optimization, boundary condition enforcement, adaptivity, and stability improvements that address complex, irregular, or evolving geometric domains.

Meshfree Methods: Progress Made after 20 Years

by Jiun-Shyan Chen

2022, Journal of Engineering Mechanics

Key finding: This paper surveys two main branches of meshfree methods—Galerkin-based weak form and collocation-based strong form—and highlights advances such as easy enforcement of essential boundary conditions, circumventing expensive... Read more

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An Adaptive Meshfree Collocation Method for Static and Dynamic Nonlinear Problems

by Gui Rong Liu

2024, Computational Methods

Key finding: This work introduces an adaptive algorithm employing meshfree strong-form collocation with radial and polynomial point interpolation methods, integrating error estimation based on solution interpolation to guide mesh node... Read more

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Study of the convergence of the Meshless Lattice BoltzmannMethod in Taylor–Green, annular channel and a porous medium flows

by Maciej Matyka and

2024, Study of the convergence of the Meshless Lattice BoltzmannMethod in Taylor–Green, annular channel and a porous medium flows

Key finding: This study assesses the convergence of the Meshless Lattice Boltzmann Method, which decouples spatial and velocity discretizations and uses radial basis function interpolation in a semi-Lagrangian framework. The authors show... Read more

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A novel node collocation technique in radial basis function collocation method applied to neutron diffusion equations

by Yiyang Luo

2024, Annals of Nuclear Energy

Key finding: By introducing ghost nodes with controlled positioning and layering outside the computational domain, this paper presents an improved radial basis function (RBF) collocation approach that enhances solution accuracy and... Read more

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2. What mesh generation and optimization strategies enhance quality, adaptivity, and computational efficiency of polygonal and polyhedral meshes for numerical simulation?

This theme covers research on generating and optimizing meshes—ranging from triangular to polyhedral and hexahedral meshes—that adapt to complex domain topologies and geometries, preserve geometric features, and improve numerical simulation accuracy and convergence. It includes algorithms for mesh quality improvement, topological constraints management, mesh simplification with minimal loss of fidelity, and parallel methods to accelerate mesh generation.

Delaunay Mesh Construction and Simplification with Feature Preserving Based on Minimal Volume Destruction

by Tongcai Wang

2023, Applied Sciences

Key finding: This paper proposes algorithms for constructing and simplifying Delaunay triangular meshes that optimize element quality and preserve geometry by introducing vertex insertion and collapse operations guided by minimal volume... Read more

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Polyhedral mesh generation and optimization for non-manifold domains

by Rao Garimella and

2016

Key finding: The authors present a method to generate polyhedral meshes for complex non-manifold geometries by computing the dual of a tetrahedral mesh and applying mesh untangling and quality improvement steps. They demonstrate that when... Read more

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Parallel Mesh Generation

by Nikos Chrisochoides

2022, Lecture Notes in Computational Science and Engineering

Key finding: This survey classifies parallel unstructured mesh generation methods according to the underlying sequential mesher (Delaunay, Advancing Front, Edge Subdivision) and the degree of coupling between subproblems, detailing... Read more

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Robust topological construction of all-hexahedral boundary layer meshes

by François Henrotte

2023, ACM Transactions on Mathematical Software

Key finding: This work introduces a robust, combinatorial integer programming approach for constructing topologically optimal all-hexahedral boundary layer meshes on complex geometries featuring arbitrary ridges and corners. It leverages... Read more

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3. How can advanced geometric parameterization and mesh morphing techniques be integrated into numerical solvers to improve shape optimization workflows?

This research focus addresses the interconnection between parametric shape description and mesh management in computational simulations, particularly techniques that directly manipulate mesh node positions ('mesh morphing') to avoid costly remeshing, improve computational robustness, and enable rapid design iterations within simulation-driven optimization frameworks.

Integration within Fluid Dynamic Solvers of an Advanced Geometric Parameterization Based on Mesh Morphing

by Ubaldo Cella

2025, Fluids

Key finding: The paper develops a procedure coupling radial basis function-based mesh morphing directly integrated into fluid dynamics solvers, bypassing the traditional remeshing and CAD model regeneration steps. This integration enables... Read more

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Generative mesh modeling

by Dieter Fellner

2015

Key finding: This research proposes representing 3D shape models functionally via a high-level mesh creation/manipulation language allowing parametric, composable shape descriptions rather than relying solely on geometric primitives or... Read more

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All papers in Meshless Methods

Meshless Methods: a Review and Computer Implementation Aspects

by Stéphane Bordas

2008, Mathematics and Computers …

Fig. 13. (a) Scheme of the visibility method and (b) scheme of the diffraction/transparency method. where w4(X) is the quartic spline and the factor a controls the amplitude of the enriched kernel function compared with the amplitude of the regular nodes. The value of a is usually set to 1. The indices c, m and p stand for cos, minus sin and plus sin, respectively. An advantage of this method is that no blending domain needs to be introduced.

Fig. 1. Discretization using meshless methods: nodes, domains of influence (circular shape).

The final characteristics of weight functions is its functional forms. The weight function should be continuous and positive in its support. For all the meshless methods that we will review in this paper, the continuity of the shape function will be determined solely by the continuity of the kernel function, for details see, e.g. [57]. For example, if the kernel function is C”, then the corresponding shape function is also C?. Oe. FA Ss WS SH RE SHR ARTA BS.

Fig. 3. Weight and shape function of the central node: (a) weight function and (b) shape function. A eS a tee A Le ie <M eae Tt can n be s seen n graphically that the MLS shape functions are indeed a partition of unity in Fig. 3 in 1D. Consider an interval 0 < x < 4 divided into four (4) equal domains. The weight and shape functions of all five (5) nodes are plotted in Fig. 3. The functions associated with the centre node are represented by a heavy line. In this example, the weight function is the quartic spline, the size of all five (5) domains of influence is 2.5. Remark that the MLS shape

Fig. 2. (a) Particle arrangement for a regular moment matrix for a linear complete MLS basis. (b) Particle arrangement for a singular moment matrix for a linear complete MLS basis. or, for the shape function ®; associated with node / at a point x:

Fig. 4. MLS shape functions and derivatives with quadratic basis. For two and three dimensions, x becomes vector x and the basis p(x) is given by (only for two dimensions):

Fig. 5. Weight and MLS shape function: (a) weight function and (b) shape function.

Fig. 6. MLS shape function first derivatives: (a) ®, and (b) @y. which is known as the Shepard function, the lowest order form of MLS shape functions. Note that the basis p is often shifted by (x — x,;)/d; since the shifting improves the conditioning of the moment matrix. A two-dimensional graphical representation of the quartic spline with the corresponding linear complete MLS shape function is shown in Fig. 5. The first spatial derivatives of the shape functions in the x- and y- directions are depicted in Fig. 6.

Fig. 7. One-dimensional problem with localized solution: comparison between EFG and enriched EFG solutions. compact support “small” relative to the domain size, to ensure that the left hand side matrix remains banded; (2) the discrete equations are modified, which complicated the implementation into existing codes. The local (extrinsic) PU enriched formulation is given by

Fig. 8. Comparison of enrichment strategies and effects of nodal support size. 2.7. Weighted residual methods

Fig. 9. String on elastic foundation: point collocation solutions. The trial and test functions in Galerkin methods are given by

2.8. Discrete equations for elastostatics

Fig. 11. Integration in Galerkin-based MMs: background mesh (left) and background structure cells (right)

Fig. 12. Undesired introduced discontinuities by the visibility method. 2.9.2. Modification of the intrinsic basis

Fig. 16. The elements of set / are nodes whose support contains point x and the crack tip Xtip- Fig. 15. The elements of set \“ are nodes whose support contains point x and cut by the crack.

Fig. 36. Geometry of the double cantilever beam. We study the cracked speciment of Fig. 36, also known as double cantilever beam (DCB). The dimensions are : follows: length L = 300 mm, height A = 100, initial crack length a = 138 mm, the load P = 100 N and mechanic properties E = 200 GPa and v = 0, 3, and we assume plane strain conditions.

Fig. 14. Two-dimensional plot of branch functions. It is clear that the first function is discontinuous through crack face.

Fig. 17. Enrichment function W with discontinuous derivative. where B* is the standard B and B“”" is the enriched B matrix:

Fig. 18. Strain computed with and without enrichment. where fis the signed distance to the discontinuity line and S° is the set of nodes whose support is cut by this line. LE Ee, Ee eS ae) ee ee ee ee a ee eee ee ee ee Pee ee ee ee Sa ee a ee in the derivative of the unknown function cannot be captured. It is clear that with this global enrichment method, one must choose smartly the enrichment function, namely this function must have local character (discontinuous derivative through a material interface, for instance) and zero elsewhere. The choice of this function is not trivial in two dimensions, especially for complex interfaces. For these cases, the local PUM-enrichment strategy works best. To model a discontinuity in the derivative (or weak discontinuity), the following approximation is used:

Fig. 19. Numerical solution with the PUM enrichment strategy.

Fig. 20. Selection of enriched particles (filled particles): (a) discontinuous enriched particles; (b) near tip enriched particles. Let us discretize the essential boundary with (m — 1) two-noded finite elements. For each element, ng, Gauss points are used. The shape functions for a two-node element are given by (/. is the length of the element)

Listing 6. Selection of enriched particles (or nodes).

Fig. 21. The Timoshenko beam. where J = D° /12 is the moment of inertia (second moment of area). The exact displacement solution for this problem The parabolic traction is given by Consider a beam of dimensions L x D, subjected to a parabolic traction at the free end as shown in Fig. 21. The beam is considered to be of unit depth and is in plane stress state. This problem was solved previously by Dolbow and Belytschko [37].

Fig. 22. 18 x 7 regular nodes distribution for the Timoshenko beam. In the computations, material properties are taken as E = 3.0 x 107, v = 0.3 and the beam dimensions are D = 12 and L = 48. The shear force is P = 1000. The regular node distribution together with the background mesh used for numerical integration of the weak form are shown in Fig. 22. In each integration cell, 4 x 4 Gauss quadrature is used. A linear basis and cubic spline weight function are used in the MLS approximation. Circular nodal support of radius 3.5 times the nodal spacing is employed.

Fig. 23. Stresses comparison: (a) shear stress; (b) normal stress. In Fig. 25, we show the error in the energy and in the displacement norm. h is the horizontal spacing between the nodes and e is the error. The convergence rate in the energy norm is 1.43.

Fig. 24. Distribution of normal stress in the beam. where /, 6 are the usual polar coordinates centered at the center of the hole.

Fig. 25. Convergence in the energy (top) and displacement (bottom) norm. The optimal rates are 1.0 and 2.0, respectively.

Fig. 26. Plate with a hole: (a) whole domain; (b) 1/4 model with irregular nodes. The cubic spline function serves as a weight function. The domain of influence for all nodes is a circle with varying radius. They are chosen such that the support is small for nodes near the hole and bigger for nodes near the edges. The essential boundary conditions are imposed with the boundary point collocation method where collocation points are coincident with nodes along the bottom and left edges. The stress 0, computed at nodes are plotted and compared to the exact solution (Fig. 28). With a coarse discretization of 99 nodes, quite accurate results are obtained.

Fig. 27. Gauss points: (a) background mesh; (b) structured cells. Consider an infinite plate containing a straight crack of length 2a and loaded by a remote uniform stress field o. Along ABCD the closed form solution in terms of polar coordinates in a reference frame (r, 8) centered at the crack tip is

Fig. 29. Infinite cracked plate under remote tension: geometry and loads.

Fig. 31. (a) Error in the energy for the mode I problem using the visibility, diffraction and transparency criterion; (b) normalized stress intensit factor vs. h. ee Ee eee eee eee The error in the energy norm is illustrated in Fig. 32(a) for some enriched methods (with crack tip enrichment), Sections 2.9.2 and 2.9.4. We have also included the results obtained with the visibility criterion in this figure. The methods that include the crack tip enrichment give more accurate results and a much better convergence rate which is expected of course. The most accurate results and the highest convergence rate of 0.94 are obtained with extrinsic MLS enrichment. However, the computational cost is higher since the enrichment is applied in the entire domain. The extrinsic PU enrichment gives a convergence rate of 0.86 and is only slightly less accurate than the extrin- sic MLS enrichment. The intrinsic PU enrichment lies in between these two results. The same observation can be made for local convergence. The fact that the SIFs can be directly obtained is a major advantage of the extrinsic MLS enrichment and probably leads to more accurate results with respect to local convergence. Nevertheless, also

Fig. 30. Computed deformed configuration (scaled).

Fig. 33. Infinite plate with circular inclusion. Fig. 35 gives a comparison of the vertical displacement fields computed by both methods.

Fig. 32. (a) Error in the energy for the mode I problem using the intrinsic and extrinsic PU enrichment, respectively, and the extrinsic MLS enrichment; (b) normalized stress intensity factor vs. h.

Fig. 34. Domain discretization with enriched nodes: (a) meshless particles; (b) finite element mesh.

Fig. 35. Vertical displacement field: (a) EFG solution and (b) XFEM solution.

Fig. 37. Stress intensity factors for a crack growth increment Aa = 5 mm at each step.

Fig. 38. Crack path (Aa = 8 mm). A perturbation of the crack direction is introduced at the crack tip: angle 4.8° and length dx = 12mm, which, physically, could be associated with the presence of a defect (void, inclusion) in the vicinity of the crack tip. [he crack propagation phenomenon is studied experimentally by Sumi et al. [96] who show the unstable nature of the crack path, which either curves downwards or upwards. The crack path is represented by a dashed line in Fig. 36. Sumi [95] solve this problem by the finite element method and Fleming et al. [50] by meshfree meth- ods (either with refined point distributions in the vicinity of the crack tip or with intrinsic enrichment of the MLS

Bnew aN ee The evolution of the stress intensity factors during propagation are shown in Fig. 37 where we see that the crack is predominantly loaded in mode I. The slightly negative SIF is sufficient to curve the crack path in the direction of the top edge of the beam. We compare the experimental results to our numerical results in Figs. 38-40 for crack propagation increments, at each step of 8, 5 and 3 mm, respectively. We note that enrichment significantly improves the results.

Fig. 41. Crack paths for various crack growth increments. Fig. 41 shows the different crack paths obtained for three values of the crack growth increment. We note that the latter should not be “too small’, otherwise, the nodal supports and J integral domains also become small which decreases the role of near-tip enrichment and poor accuracy in the stress intensity factors calculation, which exhibit spurious oscillations.

Keywords: Meshless methods; Intrinsic enrichment; Extrinsic discontinuities; Computer implementation, MATLAB

Assume that the integration is performed with background integration cells. In two dimensions, each integration cell is a four node quadrilateral element with shape functions N; and nodal coordinates xo (J = 1,..., 4). For each

The B matrix at a Gauss point gp is composed of two parts: the standard and the enriched part, where the standard part is always computed and the enriched part is only computed if in the nodes whose supports cover gp, there exist enriched nodes. This is implemented in Matlab and given in the following listing. function [sctrB] = assembly(index ,split nodes ,tip_nodes,... pos)

Comparison of vertical displacement at end of beam Table 1

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Proceedings of the First Euro-Mediterranean Workshop on Meshless Methods

by Mohammed Seaid

The intrinsic concept of meshless methods may be found in many approaches in interpolation and numerical methods for partial differential equations. Given this common concept, the aim of the Euro-Mediterranean workshop is to provide an... more

Figure 1: Magnetic Field and the Initial Condition This test problem is quasi-similar to the one used by Bonnement [1] and Sharma [2]. The axisymetric geometry is discretized with 2500 collocation points and the time scheme is the explicit Runge-Kutta 4.

Figure 3: Magnetic Flux Density and Initial Condition

The hyperbolic B-splines of order 3 possess all the desirable properties of classical polynomial B-splines, see [8]. In this paper, we limit ourselves to list some of them with the respective left and right hand side boundary hyperbolic B-splines are

seen that the agreement is very satisfactory between our results and those of the reference [8]. With an objective of validation, we consider a problem of natural convection in a square cavity linearly heated from its left vertical and bottom horizontal walls and uniformly cooled with constant temperature from its right vertical and top horizontal walls. This problem was recently investigated by Sathiyamoorthy and Chamkha [8]. The validation is conducted with a structured grid for Ra = 10° and Pr = 0.054. Qualitaive comparative results are presented in Fig. 2 in terms of isotherms (left), streamlines (middle) and local Nusselt number (right) defined as Nu, = — << . It can be x y=0 seen that the agreement is very satisfactory between our results and those of the reference [8]. In addition, other tests of validation were conducted against previous studies by El Sherbiny et al. [9] and Zhu and Y ang [10] in the case of a tall cavity with an aspect ratio A = 16 and air (Pr = 0.71) used as working fluid. Comparisons were

Figure 3: Validation of Nu against previous studies [9-10] forA = 16 and Pr = 0.71

For w C Q an open subregion of Q with positive Lesbegue measure let y,, be the restriction function defined by for all (z1,z2) € D(A) = D(A) x H'!(Q) the solution of the system (4) is expressed using the semi-group (5 (f));s0 generated byA and given by

Figure 6: The evolution of the control function. Figure 5: The desired speed gradient g“ and its reached in w

Figure 4: The desired position gradient gf and its reached in w. For w =]0, 1[ we have the figures:

This work is a continuation of the investigation of exothermic reaction fronts in porous media [1, 2, 3, 5]. It’s about a thermal frontal polymerization which is a process that involves a propagating front traveling through a monomer/initiator solution and converting monomer into polymer (Figure 1). We study the convective instability and the influence of Lewis number L, = k/d on the thermal instability, here « is the conductivity and d is the diffusivity. We firstly introduce the model and reduce it to a singular perturbation problem. Then we performe an asymptotic analysis for large activation energies that lead us to the interface problem. Finally, we discuss the stability conditions given by the linear stability analysis and present some numerical results. Figure 1: Schematic of thermal frontal polymerization

We proceed with the linear stability analysis around the basic solution of the previous interface problem using thi Fourier analysis. To linearize the jump conditions, we use the Taylor formula at the position of the reaction zone y, fo the unknown functions of the problem @ ,a, S and v. The problem is reduced to solve some algebraic relation calle dispersion relation which contain all the independent chemical parameters of the problem: /,,R,, Z, u, x. We stud numerically this relation to found both the cellular and pulsating stability boundary. In This talk we limit our self to th first case, the pulsating one, where w = iy, is under consideration. Figure | (left) shows the behavior /, in term of th wavenumber k for different value of the velocity of the front u. We note that the stability boundary is unchanged fo every values of R, or Z. Figure 1 (right) shows the variation of R, versus wavenumber k for different values of u whe: Lz! =1- Se goes to zero, we obtain the principal result found in [1] as a limiting case in the considered model in thi investigation. Figure 2: (left) /., (right) R, functions of k, for different values of u. 5. Conclusions

Figure 1: Reached position between qa, and f,

Figure 2: Reached speed between qa, and f;

In this section, we will construct an approximate of yO@) by using Taylor series expansion. According to Schoenberg- Whitney theorem (see [5]), for a given function y(@) sufficiently smooth there exists a unique hyperbolic spline s(@) = eee | 3. Hyperbolic B-splines interpolation

The exact solution is y(x) = x(1 — x) cosh(x). Result has been shown for different values of h in Table 2. References

On appliquant le théoreéme de Green-Ostrogradsky, puisque la franti¢re T est une courbe fermée lisse et reguliére on a Proof. Soient x dans Q ou dans 9°, et B, la boule de centre x et de rayon ¢, de frontiére S,. Soient Br une boule de centre 0, contenant © et x, et Sz sa frontiére. Nous appliquons la deuxiéme formule de Green dans Br M Q°. Nous obtenons On obtient une équation intégrale de Fredholm de 2éme espéce

In the TL-BM, the distribution function is approximated by the Taylor series of order 2. The previous six functions f at the points A, A, A%, As et Ag can be approximated at the second order by the Taylor series in the vicinity of A4, which gives:

Table 1: Evolution of error in any k and /.

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A robust weakly compressible SPH method and its comparison with an incompressible SPH

by Mehmet Uğur YILDIZ

2012, International Journal for Numerical Methods in Engineering

This paper presents a comparative study for the weakly compressible (WCSPH) and incompressible (ISPH) smoothed particle hydrodynamics methods by providing numerical solutions for fluid flows over an airfoil and a square obstacle. Improved... more

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Analysis of Thermoelastic Waves in a Two-Dimensional Functionally Graded Materials Domain by the Meshless Local Petrov-Galerkin (MLPG) Method

by Stéphane Bordas

This contribution focuses on the simulation of two-dimensional elastic wave propagation in functionally graded solids and structures. Gradient volume fractions of the constituent materials are assumed to obey the power law function of... more

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Meshless natural neighbor Galerkin method for the bending and vibration analysis of composite plates

by Madhukar Somireddy

In the present work we present a meshless natural neighbor Galerkin method for the bending and vibration analysis of plates and laminates. The method has distinct advantages of geometric flexibility of meshless method. The compact support... more

Fig. 1. (a) First order Voronoi diagram and Delaunay triangulation of a set of points. (b) First order Voronoi diagram with additional point x. The shaded area A.g.r divided b the area Agica of the Voronoi cell V(x) defines the Sibson interpolant N9(x). (c) Sibson interpolant. To illustrate the geometric interpretation of C°-continuous nat- ural element shape functions, consider the Voronoi diagram in Fig. 1(b). According to Eq. (1), the C°-continuous natural element shape function N?(x) with respect to node p, is defined as the ratio of the shaded area of overlap of the Voronoi cell V(p,) to the total area of the Voronoi cell of V(x), ie, N9(x) =A(V(p7) V(x) /V(x ) =A caet/Aabca. It thus takes the interpretation of the area ratio of the polygons {cdef} and {abcd}. The C°-continuous natural neighbor interpolants are the Sibsons interpolants, based on the first order Voronoi diagram. Consider a set of nodes V = {p,,po,..-,Pu} € Ri. The first order Voronoi dia- gram V(N) of the set VV is a subdivision of the Euclidian space R@ into convex regions V(p,) = {x ¢ R¢: ||x—py|| < |x — p)|| WATS called Voronoi cells. The above definition states that any point in the Voronoi cell V(p,) is closer to node p, than to any other node Pp, € N. Fig. 1(a) shows the first-order Voronoi diagram and the cor- responding dual Delaunay triangulation for an example consisting of eight node points. Fig. 1(b) illustrates how the natural neighbor interpolant is constructed. For the set of VV nodes considered, the Voronoi diagram is first computed. Now, to interpolate the function value at an arbitrary point x, we insert the point x in the set of nodes

Fig. 2. (a) Shape function derivatives along x. (b) Shape function derivatives along y.

The constitutive relation for a laminated composite plate can be written in lamina coordinates (x,y,z) and the in-plane stress strain relation for the kth lamina is given as

Non-dimensional central deflection and in-plane stresses of a simply supported orthotropic plate with a/b =1 under uniformly distributed loading with G,, is calculated at (x,y,Z) = (a/2, b/2,h/2).

Non-dimensional central deflection and in-plane stresses of a simply supported symmetric (0°/90°/90°/0°) laminated plate with a/b = 1 under uniformly distributed loading wit Gyy is calculated at (x,y,z) = (a/2,b/2,h/2).

Non-dimensional central deflection and in-plane normal stress of a simply supported isotropic plate with a/b = 1 under uniformly distributed loading

Non- dimensional fundamental frequency (@ = wh,/2(1 + v)p/E) of a simply supported isotropic rectangular plate with a/b = 2 and different a/h ratios.

Non-dimensional fundamental frequency (@ = (cob? /h) \/p/E2) of a simply supported (0/90/90/0) laminated plate with a/b = 1 and different a/h ratios.The material properties of composite plate are E;/E, = 40, Gy2 = Gi3 = 0.6E2, G23 = 0.5E2, Vi2 = V13 = V23 = 0.25. Note HSDT1, corresponds to displacements u, v,w are cubic variations of z. HSDT2, corresponds to displacements u, v are only cubic variation of z and w(x, y).

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ENGINEERING PHYSICS AND MATHEMATICS RBF-DQ method for solving non-linear differential equations of Lane-Emden type

by Soleiman Hashemi

The Lane-Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. In this paper, a new numerical method is applied to investigate some well-known classes of Lane-Emden... more

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A simple and robust three-dimensional cracking-particle method without enrichment

by Stéphane Bordas

2010

A new robust and efficient approach for modeling discrete cracks in meshfree methods is described. The method is motivated by the cracking-particle method (Rabczuk T., Belytschko T., International Journal for Numerical Methods in... more

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Coupled Fluid-Solid Geomechanical Modeling of Multiple Hydraulic Fractures Interacting with Natural Fractures and the Resulting Proppant Distribution

by Sam Raymond

The application of a new Material Point Method (MPM) approach to model the proppant distribution in a reservoir where hydraulic fractures interact with natural fractures is presented and validated with an Eagle Ford well. The new MPM... more

Figure 1—Schematic of elementary tests involving one hydraulic and one natural fracture. To understand the behavior of fluid and solid interaction within a fracture, as well as the following propagation, a number of elementary tests using the data described in Aimene and Nairn (2014) were performed. These involved a single hydraulic and a single natural fracture, placed within the material and oriented at various degrees from the horizontal axis. Lateral compressive stress is applied on the sample reproducing the regional stress whose anisotropy is equal to one. A diagram of this is shown in Fig. | of the geometry of the hydraulic and natural fracture.

Figure 2—(A) Elementary test of a 30° natural fracture: a) proppant entering the hydraulic fracture. b) intersection of hydraulic and natural fracture and proppant entry into the natural fracture. c) proppant filled natural fracture and begins to propagate. d) further propagation of natural fracture with proppant flow. (B) Proppant concentration during the simulation. yf the fracture whose material has a critical value for fracture propagation. The proppant, represented as he fluid material points, pumped into the hydraulic fracture was pushed further into the fracture and ‘ventually propagated into the natural fracture. Four simulation times for the three elementary test results re shown in Figs 2-4 for the 30°, 60°, and 90° cases, respectively. Fig. 4 represents a zoom of the las imulation for the 60° case where the red points represent the latest distribution of the proppant. In al hree cases the proppant successfully opened the hydraulic fracture and propagated it to intersect with the learby natural fracture. Once the two fractures had crossed, the proppant entered the natural fracture and yegan imparting pressure on the natural fracture. The increase in fluid pumping volume resulted in the atural fractures propagating with the proppant flowing along with it.

Figure 3—(A) Elementary test of a 60° natural fracture: a) proppant entering the hydraulic fracture. b) intersection of hydraulic and natural fracture and proppant entry into the natural fracture. c) proppant filled natural fracture and begins to propagate. d) further propagation of natural fracture with proppant flow.propagate. d) further propagation of natural fracture with proppant flow. (B) Proppant concentration during the simulation.

Figure 4—Zoom around the hydraulic and natural fractures showing the asymmetric proppant distribution

Figure 5—(A) Elementary test of a 90° natural fracture: a) proppant entering the hydraulic fracture. b) intersection of hydraulic and natural fracture and proppant entry into the natural fracture. c) proppant filled natural fracture and begins to propagate. d) further propagation of natural fracture with proppant flow.(B) Graph of proppant concentration during the simulation. Notice the inability of the proppant to open new patheways causing a screen out open fracture area and the number of particles within each fracture to calculate a concentration of proppant within a unit area of open fracture. Figs 2B, 3B, 5B show the proppant concentration plot. Additionally, key features of the proppant concentration curves related to leak-off are identified for the 4 distinct steps of the simulation: (a) fluid entering the hydraulic fracture, (b) fluid propagating and intersecting the hydraulic and natural fracture, (c) fluid entering the natural fracture, and finally (d) fluid filling the natural fracture and propagating the natural fracture.

Figure 6—(A) Evolution of proppant placement in a complex fracture network of configuration with favorable natural fracture orientations. (B) Total proppant concentration versus time.

Figure 7—Zoom on the last proppant distribution from Fig. 6 showing the ability of the proppant to propagate farther when encountering the fracture set oriented N40E. The fracture set oriented N10E, seems to prevent the proppant placement. placement. Fig. 7 shows a zoom of the proppant distribution at the end of the simulation showing the successful placement especially in the upper natural fracture that has a favorable fracture orientation.

Figure 8—(A) Evolution of proppant placement in a complex fracture network of configuration with unfavorable natural fracture orientations. (B) Total proppant concentration versus time.

ire 9—(A) The Equivalent Fracture Model (EFM) around the last three stages of the Eagle Ford well described in Ouenes et al. 2015 Main proppant flow pathways showing the best asymmetric half length at completion stage #7.

Figure 10—(A) shows a proppant simulation involving three hydraulic fractures at the heel of an Eagle Ford well and the resulting proppant distribution at four key time steps, (B) evolution of the proppant concentration vs. time as the proppant invades and avoids different natural fractures

Figure 11—(A) Slurry concentration measured during the refrac of the same considered Eagle Ford well showing similar features as those shown in the simulated proppant concentration.) Moving average of the proppant concentration curve of Fig. 11B showing some of the key features related to the impact of natural fractures on proppant concentration.

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Adaptive Natural Element Method for Analysis of Plates and Laminates_ICCMS12

by Madhukar Somireddy

In this paper, a natural element method (NEM) is employed for the analysis of plates and laminates. The displacement field and strain field of plate are based on Reissner-Mindlin plate theory. Sibson interpolation [4] based on natural... more

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Three-Dimensional Crack Initiation, Propagation, Branching and Junction In Non-Linear Materials by An Extended Meshfree Method Without Asymptotic Enrichment

by Stéphane Bordas

2008, Engineering Fracture Mechanics

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Two-dimensional meshless solution of the non-linear convection diffusion reaction equation by the Local Hermitian Interpolation method

by Henry Power and

2013

"A meshless numerical scheme is developed for solving a generic version of the non-linear convection-diﬀusion-reaction equation in two-dim-ensional domains. The Local Hermitian Interpolation (LHI) method is employed for the spatial... more

To implement the LHI method it is necessary to cover the solution domain with two sets of scattered points as shown in Figure 1. The points are the nodes where the solution and the PDE will be two sets o enforced by interpolation; nodes and which is t The stencil (big circles in ] are the so PDE centres. he subdomain w hence they are called, respectively, solution Each solution node has an associated stencil, here a specific interpolation function is valid. Figure 1) contains three kinds of points which ution nodes, the PDE centres and the boundary points when the stencil intersects the domain boundary. For more information about the effect of the stencil nodal distribution on the performance of the method, see [24],[25].

[wo-dimensional meshless solution of the non-linear convection diffusion reaction equation So far a local interpolation equation is obtained for the stencil k, considering the coefficient a“) in terms of the variable values at the solution centres included in the stencil. Then the original operator of the problem N,, is applied to the dependent variable expressed as (7) and the resulting equation is evaluated at the central solution point x;,, to obtain the non-linear expression (9) which relates the unknown variable values contained inside the stencil k.

global convergence rate and, in consequence, the fewest number of global iterations for R < 100. For Re = 1000 the number of global iteration in the NRIINJ strategy is the highest due to the ill-conditioning of the local interpolation matrices which causes an inaccurate response to the small perturbations needed to compute the numerical Jacobian matrix. The double iteration strategies perform better than the single iteration one considering the CPU time spending and the stability of the method as the parameter R is increased. As shown in Figures (3) b. and (4) b. the convergence rate of the double iteration strategies are almost the same while in the NRIINJ case the rate is higher for R = 100 and smaller for R = 1000. As is expected, the Picard iteration strategy gives the lowest convergence rate. Picard iteration strategy gives the lowest convergence rate.

Table 2: One-dimensional Burgers’ equation solution by several non-linear solvers

Table 3: Two-dimensional Richards’ equation solution by several non-linear solvers When a = 3 the NRIINJ strategy is unable to solve the problem whatever the initial values are and the HAM2 spends approximately 4 times the CPU time required by the Picard iteration strategy. In case of NRIINJ, the ill-conditioning of the interpolation matrices causes poor accuracy in the numerical Jacobian computation. The equivalence among the attained solutions is verified in the Figure 6 a. which shows the absolute error at line x2 = 0.5 for a = 3. The Figure 6 b. allows to have a qualitatively idea of the high global convergence rates which the double iteration strategies present with respect to the Picard iterative scheme. The Figure 6 c. shows the slow convergence rate of the HAM strategy in the solution of the non-linear equation system while the double iteration Newton Raphson schemes present

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Analysis of rectangular square plates by the mixed Meshless Local Petrov-Galerkin (MLPG) approach

by Tomislav Jarak

A new mixed meshless formulation based on the interpolation of both strains and displacements has been proposed for the analysis of plate deformation responses. Kinematics of a three dimensional solid is adopted and discretization is... more

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NUMERICAL MODELLING OF WAX DEPOSITION USING SPH

by Mahdi Haddad and

Phase change and deposition of solid particles in liquid flows are undesirable in some natural and industrial processes and can be hazardous in some cases. Considering the difficulties involved in petroleum exploitation in deep waters,... more

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On the numerical implementation of the higher-order strain gradient-dependent plasticity theory and its non-classical boundary conditions

by Rashid Abu Al-Rub

2015, Finite Elements in Analysis and Design

The higher-order gradient plasticity theory is successful in explaining the size effects encountered at the micron and submicron length scale. Due to the incorporation of spatial gradients of one or more internal variables in these... more

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Simulating 3-D Single Gas Bubble Growth in a Polymer Melt Using Multi-Phase SPH

by Burak Kaan CIRPICI

Modelling of the expansion of 3-D single bubble using a multi-phase model has been developed for GIVE APPLICATION AREA with the potential of a meshless numerical simulation method, Smoothed Particle Hydrodynamics (SPH), and the... more

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robust weakly compressible SPH method and its comparison with an incompressible SPH

by M. S. Shadloo

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An Adaptive–Hybrid Meshfree Approximation Method

by Leevan Ling

SUMMARY It is now commonly agreed that the global radial basis functions method is an attractive approach for approximating smooth functions. This superiority does not come free; one must find ways to circumvent the associated problem of... more

Figure 1. (Left) Generic labeling for each individual cell; split cell if the principle P(C) > 7. (Right) Schematic illustration of cells splitting in 2D and 3D views.

Figure 2. Effects of a and 6 in the principle function Py if when applying ANR to f(x,y) = (a + y)/10 + exp(10(—(a + 0. 1) —y”)).

Figure 3. Non-nested level structures on the ANR output refined nodes. Approximation s1 to f =exp(x+y)/5 + exp(—1000((« — 3/4)? + y”)) obtained by assigned nonzero weights to the first two node-levels only.

Figure 4. Example 4.1: RMS errors against different numbers of data cites and degrees of freedom in the numerical approximations, respectively, given in lines and markers. (Left) various MQ shape parameters c with fixed a = 0; (Right) various interior principle parameters a with fixed c = 1.

Figure 5. Example 4.1: (Left) Numerical approximation to the smooth-octopus and (Right) associated residual function.

Figure 6. Example 4.2: (Left) a numerical approximation to the PEAKS function; (Right) relative L?(Q) errors €2 with different (c, a) against the number of data sites N.

Figure 8. Example 4.3: (Left) Number of nonzero entries in State II against the numbers of data sites; (Right) sparsity pattern of the collocation matrix in Stage I with 7 = 1/14”.

Figure 9. Example 4.3: (Left) a numerical approximation to the elongated near-singularities; (Right relative L?(Q) errors €2 with different (c, ~) against the number of data sites N

Figure 10. Example 4.4: (Left) a numerical approximation to the boundary layers; (Right) relative L?(Q) errors €2 with different (c, a) against the number of data sites N.

Figure 11. Example 4.5: (Left) a numerical approximation to the shock formation; (Right) relative L?(Q) errors € with different (c, a) against the number of data sites N.

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MESH SIMPLIFICATION VIA A VOLUME COST MEASURE

by International Journal of Computer Graphics & Animation (IJCGA)

2015, International Journal of Computer Graphics & Animation (IJCGA)

We develop a polygonal mesh simplification algorithm based on a novel analysis of the mesh geometry. Particularly, we propose first a characterization of vertices as hyperbolic or non-hyperbolic depending upon their discrete local... more

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A new weight-function enrichment in meshless methods for multiple cracks in linear elasticity

by Vito Tagarielli

2012

A new enriched weight function for meshless methods is proposed for the numerical treatment of multiple arbitrary cracks in two dimensions. The main novelty consists in modifying the weight function with an intrinsic enrichment which is... more

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On Shear Locking in MLPG Solid-Shell Approach

by Tomislav Jarak

A solid-shell MLPG approach for the numerical analysis of plates and shells is presented. A special attention is devoted to the transversal shear locking effect that appears in the structure thin limit. The theoretical origins of shear... more

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A simple error indicator for meshfree methods based on natural neighbors

by David Ryckelynck

2006, Computers & Structures

The main aim of this paper is the development of a refinement procedure able to operate in the context of the constrained natural element method (C-NEM). The C-NEM was proposed by the authors in a former work [Yvonnet J, Ryckelynck D, Lorong P, Chinesta F. A new extension of the natural element method for non-convex and discontinuous domains: the constrained natural element method (C-NEM). Int J Numer Methods Eng 2004;60:1451-1474] and its main meshless features, that allow to describe large domain changes as well as to handle fixed or moving discontinuities, were analyzed in [Yvonnet J, Chinesta F, Lorong P, Rynckelynck D. The constrained natural element method (C-NEM) for treating thermal models involving moving interfaces. Int J Thermal Sci 2005;44:559-569; Yvonnet J, Lorong P, Ryckelynck D, Chinesta F. Simulating dynamic thermo-elastoplasticity in large transformations with adaptive refinement in the natural element method: application to shear banding. Int J Forming Process 2005;8:347-363]. Sometimes, in order to improve the interpolation accuracy for describing boundary layers or an anisotropic behavior, new nodes must be added, removed or repositioned. The interpolation in the vast majority of meshless techniques is free of mesh quality requirement. Thus, introduction, elimination or repositioning of nodes is a trivial task, because no geometrical restrictions exist. In this way, nodes can be added without geometrical checks in the regions where the solution must be improved (identified by using an appropriate error indicator). For this purpose, in this paper an a posteriori error indicator will be proposed and tested in some linear elastostatic problems benchmarks involving different levels of difficulty (stress concentration, solution singularities, . . .) all of them with a known exact solution. The computational implementation of this error indicator is very simple, and when it is used in tandem with an efficient refinement procedure, which makes use of the meshless features of the C-NEM, provides an accurate adaptation procedure, specially appropriate in the C-NEM framework.

Fig. 8. Effectivity index associated with the NNI and the NN2 error indicators for the plane stress plate problem.

Fig. 10. Energy norm of the errors in the plate with a hole problem.

Fig. 9. (a) Square plate with a central hole. (b) Computational domain and boundary conditions.

To extend this solution to the whole domain, we impose on the square plate boundary the exact traction: Fy=o,,n, where we assume L=1mm, a=L/2mm and K;=1. The constrained Voronoi diagram allow us to introduce easily and accurately the presence of the crack as well as continuity around the crack tip without any special treat- ment due to the salient features of the C-NEM shape func- tions support [42]. Fig. 11. Effectivity index in the plate with a hole problem.

Fig. 13. (a) Mode-I crack problem geometry. (b) Exact boundary conditions. Fig. 12. Adaptive refinement in the plate with a hole problem.

Fig. 1. Construction of the Sibson shape functions. (volume in 3D) of the Voronoi cell associated with the point x:

‘Introducing now the natural element approximation of the displacement field according to Eq. (6) we obtain:

Fig. 2. Integration domains for the SCNI procedure: (a) intersection between the Voronoi diagram and the domain closure and (b) constrained and bounded Voronoi diagram.

Fig. 3. Computing errors in each Voronoi cell. J. Yvonnet et al. | Computers and Structures xxx (2006) xxx-xxx

The strategy depicted in Fig. 4(a) has been considered by several authors [41,29] and is suitable to refine irregular clouds of nodes. Starting from an initial cloud of nodes, we first construct the associated Voronoi diagram. If the condition (43) is valid in a given Voronoi cell, then new nodes are added on the vertices of the cell. This operation is repeated for all cells. Then We reconstruct the Voronoi diagram of the new cloud of nodes, and so on until the glo- Fig. 5. (a) Square plate with plane stresses. (b) Constrained Voronoi cells. A square plate of length L is subjected to the following system of loads (as depicted in Fig. 5):

Fig. 4. Strategies of nodal refinement based on the Voronoi cells. (0): additional nodes.

Fig. 6. Convergence of the exact and computed stresses for the plane stress plate problem. A traction load is applied on a square plate as depicted in Fig. 9 under plane stress conditions. The exact stress solution for an infinite plate subjected to a unit traction on the boundaries whose unit normal vector coincides with the x, direction, is given by [38]

Fig. 7. Convergence of the different errors in a plane stress plate problem.

Fig. 14. Effectivity index in the mode-I crack problem.

Fig. 15. Refinement procedure in the mode-I crack problem. In this paper a refinement strategy able to operate with accuracy in the C-NEM framework was presented and has been tested in some linear elastostatic problems. A simple a posteriori error indicator based on recovery stress in tan- dem with stabilized nodal integration has been proposed and tested in some benchmarks problems involving differ- ont levels of difficulty (stress concentration, solution singu- arity,...) all of them with a known exact solution. The indicator exhibit asymptotically exact error estimation in problems with different level of stress concentrations. In problems involving singularities, the effectivity is reason- able and constant, which makes it a useful tool for nodal refinement in the context of the natural element method. Different adaptation procedures have been performed suc- cessfully, proving the utility of the procedure.

Effectivity indexes for the different problems Table 1

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A node enrichment adaptive refinement in Discrete Least Squares Meshless method for solution of elasticity problems

by Mohammad Afshar

2012, Engineering Analysis with Boundary Elements

In this paper, an adaptive refinement procedure is proposed to be used with Discrete Least Squares Meshless (DLSM) method to obtain accurate solution of planar elasticity problems. DLSM method is a newly introduced meshless method based... more

Fig 1: Voronoi diagram (¢@: Initial nodes, A: Added nodes)

Fig.2: An adaptive h-refinement procedure based on error estimation

Fig. 6: The vertical displacement of the cantilever beam under the end load

Fig. 9: The convergence rate in the cantilever beam under the end load

b) Third adaptive nodal configuration (345 nodes)

Fig.15: The normal stress o,,, a) exact solution and b) at final adaptive configuration (857 nodes)

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An h-adaptive method in the generalized finite differences

by Francisco Ureña

2003, Computer Methods in Applied Mechanics and Engineering

This paper describes an h-adaptive method in generalized finite difference (GFD) to solve second-order partial differential equations. These equations representing the behaviour of many physical processes. The explicit difference formulae... more

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Coupling of the mesh-free finite cloud method with the boundary element method: a collocation approach

by Glaucio H. Paulino

2003, Computer methods in applied mechanics …

Meshless and mesh-based methods are among the tools frequently applied in the numerical treatment of partial differential equations (PDEs). This paper presents a coupling of the meshless finite cloud method (FCM) and the standard... more

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Cohesive-zone models, higher-order continuum theories and reliability methods for computational failure analysis

by Harm Askes

2004, International Journal for Numerical Methods in Engineering

A concise overview is given of various numerical methods that can be used to analyse localization and failure in engineering materials. The importance of the cohesive-zone approach is emphasized and various ways to incorporate the... more

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Benefits of Using Non-consolidated Domain Influence in Meshless Local Petrov-galerkin (Mlpg) Method for Solving Lefm Problems

by Ameen Aledani

http://www.hrpub.org, Universal Journal of Mechanical Engineering

This paper presents an efficient meshless method in the formulation of the weak form of local Petrov-Galerkin method MLPG. The formulation is carried out by using an elliptic domain rather than conventional isotropic domain of influence.... more

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Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment

by Timon Rabczuk and

2007, Engineering Fracture Mechanics

This paper presents a three-dimensional, extrinsically enriched meshfree method for initiation, branching, growth and coalescence of an arbitrary number of cracks in non-linear solids including large deformations, for statics and... more

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A simple and robust three-dimensional cracking-particle method without enrichment

by Nguyen Xuan Hung

2010, Computer Methods in …

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Analysis of Shell Deformation Responses by the Meshless Local Petrov-Galerkin (MLPG) Approach

by Tomislav Jarak

A meshless computational method based on the local Petrov-Galerkin approach for the analysis of shell structures is presented. A concept of a three dimensional solid, allowing the use of completely 3-D constitutive models, is applied.... more

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Technical Article Overview of Meshless Methods

by Boddula Swathi

Electromagnetics is also given.

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Meshless Local Petrov-Galerkin (MLPG) Formulation for Analysis of Thick Plates

by Tomislav Jarak

An efficient meshless formulation based on the Local Petrov-Galerkin approach for the analysis of shear deformable thick plates is presented. Using the kinematics of a three-dimensional continuum, the local symmetric weak form of the... more

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A meshless cohesive segments method for crack initiation and propagation in composites

by Ettore Barbieri

2011, Applied Composite Materials

A modeling method aimed at eliminating the need of explicit crack representation in bi-dimensional structures is presented for the simulation of the initiation and subsequent propagation within composite materials. This is achieved by... more

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Approximation of multivariate functions and evaluation of particular solutions using Chebyshev polynomial and trigonometric basis functions

by Sergiy Reutskiy

2006, International Journal for Numerical Methods in Engineering

A two-stage numerical procedure using Chebyshev polynomials and trigonometric functions is proposed to approximate the source term of a given partial differential equation. The purpose of such numerical schemes is crucial for the... more

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Influence of several factors in the generalized finite difference method

by Francisco Ureña

2001, Applied Mathematical Modelling

In this paper it is possible to appreciate the great eciency of the generalized ®nite dierence method (GFD), that is to say with an irregular arrangements of nodes, to solve second-order partial dierential equations which represent the... more

If the norm B(u) is minimized with respect to Ou /Ox, Ouo/Oy, C7u0/Ox", O’uo /Ov’, Ouo /OxOy and if these derivatives are indicated as, «, f,y, 6,6, respectively, [19] we then obtain: The above system may be expressed in the following matrix form:

This is then solved in two stages, the first of which being:

On including the explicit expressions for the values of the partial derivatives (16) in the initial equation (la) this then gives:

Eight segment criterion: nodes “1,2,3,4,5,6,7,8” Four quadrant criterion: nodes “1,a,3,b,4,5,7,c” Fig. 1. Distance criterion.

Fig. 4. Influence of the number of nodes on the star. The equations are solved in a square domain of unit side, except for cloud 4, and in all cases a weighting function (28) has been used. Table 1 Influence of the criterion of selection on nodes

4.3. Influence of the weighting function Table 2 shows the degree of error obtained when varying the weighting function in order to analyse the influence of the weighting function. First, solving the Laplace equation for clouds 4, 5 and 6; and then solving Eq. (34) for cloud 7, in a unit side squared domain, except for cloud 4, which is a “L”’ shaped domain (Fig. 5). The same four quadrant criterion of selection of nodes has been employed for each cloud of points together with potential, exponential and spline weighting functions.

Fig. 6. Convergence test for the solution error. Influence of the weighting functions

Fig. 7. Covergence test for the partial derivatives error.

Temperature and residual errors in steady-state problems using GFDM

Temperature and gradients errors in transient problems using GFDM Table 4

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A grid-free upwind relaxation scheme for inviscid compressible flows

by Raghurama Rao Suswaram

A new Grid-free Upwind Relaxation Scheme for simulating inviscid compressible flows is presented in this paper. The non-linear conservation equations are converted to linear convection equations with non-linear source terms by using a... more

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Analysis of Thermoelastic Waves in a Two-Dimensional Functionally Graded Materials Domain by the Meshless Local Petrov-Galerkin (MLPG) Method

by akbar bagri

2010

Abstract: This contribution focuses on the simulation of two-dimensional elastic wave propagation in functionally graded solids and structures. Gradient volume fractions of the constituent materials are assumed to obey the power law... more

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Analysis of cornea curvature using radial basis functions - Part I: Methodology

by Graham W Griffiths

2016

We discuss the solution of cornea curvature using a meshless method based on radial basis functions (RBFs). A full two-dimensional nonlinear thin membrane partial differential equation (PDE) model is introduced and solved using the... more

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Mixed discrete least squares meshless method for planar elasticity problems using regular and irregular nodal distributions

by Mohammad Afshar

2012, Engineering Analysis with Boundary Elements

A Mixed Discrete Least Square Meshless (MDLSM) method is proposed for the solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses... more

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Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations

by Bozidar Sarler

2012, Applied Mathematical Modelling

This paper examines the numerical solution of the transient nonlinear coupled Burgers' equations by a Local Radial Basis Functions Collocation Method (LRBFCM) for large values of Reynolds number (Re) up to 10 3 . The LRBFCM belongs to a... more

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An h-adaptive method in the generalized finite differences

by rafael alvarez

2003, Computer Methods in Applied Mechanics and Engineering

Fig. 2. Example of adding four new nodes to a star. In the following numerical results we compare with the analytical solution in each case and using a value limit of global error for the results of the global error obtained using formula (18).

Fig. 3. The successive clouds of points when the h-adaptive method is applied.

Fig. 1. Three examples of clouds of different density. with formula (18) is given when solving Laplace equation in three different cases, by using GFD formulae including second- or third-order of the Taylor series. Let us then consider three different cases, solving the Laplace equation with Dirichlet boundary conditions, and using in all cases the weighting function, exp(—5(x” +”). q ce e .# ° 1 . nr eo, ef 4 7 4 nr e . “7

Fig. 4. Global error reduction when the h-adaptive method is applied to solve Laplace equation. Fig. 4 shows the reduction in the global error calculated according formula (18), when the h-adaptive method is applied. Application of the adaptive method to solve the equation

Fig. 5. The successive clouds of points when the h-adaptive method is applied.

Fig. 6. Global error reduction when the h-adaptive method is applied to solve the hyperbolic differential equation (21).

J.J. Benito et al. | Comput. Methods Appl. Mech. Engrg. 192 (2003) 735-759 Application of the adaptive method to solve the equation

J.J. Benito et al. | Comput. Methods Appl. Mech. Engrg. 192 (2003) 735-759 adaptive method to cioud 11 with dpa equals to V.UZ5 dmax and an error limit OF V.UUUL. Fig. 11 shows how the solution global error (18) decreases by the application of the adaptive method adding a few number of new nodes in each cloud. As can be seen in Fig. 10, the application of the method concentrates the new nodes at the sharper corners, which is where the generalized differences formulae gives bigger errors. In this case, the corners also coincide with the high gradients area of the exact solution (26). This is the reason why the error decreases quickly in Fig. 11. a gee 6 ae

In all cases considered the h-adaptive method may be seen to provide an error reducing procedure. However, it is necessary to take into account that the error limit value for each step of the adaptive method, is conditioned by the fact that it is larger than the average of the estimated error of nodes. Moreover, it is recommended that the number of inner nodes that overcome this error limit should not be greater than 20% of the inner number of nodes, as it is better to apply the adaptive method several times and to fix J.J. Benito et al. | Comput. Methods Appl. Mech. Engrg. 192 (2003) 735-759

J.J. Benito et al. | Comput. Methods Appl. Mech. Engrg. 192 (2003) 735-759 progressively smaller error limits, than to impose a very small error limit at the outset. Then more balanced clouds of nodes are obtained at the end of the adaptive process. The computer program automatically provides the indicator error value for each node, the number of nodes with an error over the error limit and the number of new nodes to be added, in such a way that is possible to adjust the error limit, before starting the new stage, in order to suit the sed conditions.

J.J. Benito et al. | Comput. Methods Appl. Mech. Engrg. 192 (2003) 735-759 5.3. Influence of the minimum distance In this section, the adaptive method is used to solve a second-order partial differential equation with the objective of checking the minimum distance influence (dpa) in the clouds refinements obtained with the h- adaptive algorithm. Different possibilities of adding nodes procedure are explored in this example de- pending of the control parameters used. ee ee eee NT ce og Se ge ee es Be ce Be

Fig. 16. Different possibilities of adaptivity of clouds of points. J.J. Benito et al. | Comput. Methods Appl. Mech. Engrg. 192 (2003) 735-759

J.J. Benito et al. | Comput. Methods Appl. Mech. Engrg. 192 (2003) 735-759

Fig. 22. Beam with irregular distribution of nodes, Lw = w" = f(x), f(x) = 1/2 (? — 1), WY =f" = Then adding both Taylor approximations and operating

Fig. 25. (Error indicator) (cloud 9). J.J. Benito et al. | Comput. Methods Appl. Mech. Engrg. 192 (2003) 735-759

Fig. 30. (Error indicator) (cloud 17). Fig. 31. (Error indicator) (cloud 18).

On solving second-order partial differential equations by GFDM two types of errors are introduced in the solution: (a) the error due to use of GFDM, and (b) the error due to truncation of terms higher than the second-order in the generalized finite differences formulae. The error ic evaliated neino the following olohal error foarmila:

Global errors for different possibilities of adaptivity of clouds of points limit is decreased to 0.001 in a second step, no nodes are added. Then if the error limit is decreased again, the irregular cloud 17 (92 nodes), in Fig. 16, is obtained using as error limit 0.0005 in the second step. Clouds 13-17 show how “bald patches” may appear in certain areas. For example in order to eliminate a blank area, five nodes are added by hand in cloud 17. The irregular cloud 18 (105 nodes), in Fig. 16, is then obtained after applying the algorithm to the modified cloud 17 (92 + 5 nodes) with dpa equal 0.05 dmax and an error limit of 0.00023.

Fig. 20. (Abs. error) (cloud 18). and the weight function is the quartic spline function

Then adding both Taylor approximations and operating

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Meshless methods: a review and computer implementation aspects

by TOUSEEF AHMAD

2008, Mathematics and Computers …

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Imposing boundary conditions in the meshless local Petrov–Galerkin method

by Alexandre Ramos Fonseca

2008, IET Science, Measurement & Technology

A particular meshless method, named meshless local Petrov -Galerkin is investigated. To treat the essential boundary condition problem, an alternative approach is proposed. The basic idea is to merge the best features of two different... more

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A nearest-nodes finite element method with local multivariate Lagrange interpolation

by Yunhua Luo

2008, Finite Elements in Analysis and Design

In the proposed nearest-nodes finite element method (NN-FEM), finite elements are used only for numerical integration; while shape functions are constructed in a similar way as in meshless methods, i.e. by using a set of nodes that are... more

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Provided for non-commercial research and education use. Not for reproduction, distribution or commercial use.

and so is the strain energy density. After numerical integration is conducted, the strain energy in an element is also represented by nodal unknowns. The Hessian matrix of the strain energy with re- spect to nodal unknowns is the element stiffness matrix. Therefore, in calculating an element stiffness matrix, what we essentially need are the estimations of displacements and their derivatives at quadra- ture points over the element. Based on interpolation theory, with a given set of nodes and corresponding function values at the nodes, the best estimation for the function at a concerned point is obtained by interpolation using a subset of nodes that are the nearest to that point.

Fig. 4. Selected nodes for constructing Lagrange shape functions.

Fig. 3. Selection of nearest nodes for a quadrature point near a crack: (a) an existing crack; (b) a new crack.

From Eq. (6), the first-order derivatives of the Lagrange shape func- tions are calculated as Higher order derivatives can be obtained by the chain rule, but in this paper we only need approximate function and its first-order derivatives. Continuity and reproducing properties of the multivari- ate Lagrange interpolation method has been studied in [34]. Two sets of shape functions constructed using different number of nodes as shown in Figs. 5(a) and 6 are plotted in Figs. 5 and 7. The shape functions were obtained in the way described in Section 2.1 using a set of given nodes. From the figures, it can be observed that, both the shape functions and their first-order derivatives are continuous.

Fig. 5. Shape functions constructed from four nodes: (a) node distributions; (b) ¢; (c) 0;/0x; (d) 0¢,/Oy.

Fig. 6. Distribution of nine nodes (‘e’—¢,; ‘o’—2; ‘K’—¢3).

Fig. 7. Shape functions and their first-order derivatives constructed from nine nodes: (a) $15 (b) by: (c) 33 (d) 0G 4/Ox; (€) Ob2/Ox; (f) O3/0x; (g) 0p, /Oy; (h) Od2/Oy; (i) 0f3/0y.

Fig. 9. h-Convergence (BO—base order in MLS, PO—polynomial order in NN-FEM).

Fig. 10. Regular and distorted meshes: (a) regular mesh; (b) distorted mesh.

Fig. 11. A plate with an edge crack: (a) boundary and loading conditions; (b) undeformed configuration. Transverse displacement at the free end of cantilever Table 1

Fig. 12. Deformed configuration and effective stress: (a) deformed configuration; (b) effective stress.

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Efficiency Improvement Application in Computer Networks Installed in Historical Places

by Ahmet N U S R E T ÖZALP

2015, Academic Platform Journal of Engineering and Science

In today’s information society, reaching information fast and safely is the primary purpose. Computer networks are the mostly used way to share information. The increase in access and sharing demands has caused the users to look for new... more

n the study, network devices seen in Table 2 have been used. Considering the performance, economical and dipole antenna compatible devices have been preferred. While choosing the devices, what was taken into consideration was their being able to be found easily in market conditions. In this study, firstly, the ocation of the access points has been determined. In a four historical building, where above-wall cabling was not possible, mesh network topology has been focused on not to damage the historic fabric. Related to the location gateway, a network with dipole antennas has been designed in a way in which access points will be able to see one another at appropriate dipole angles. In the places where possible, the existing wireless network has been supported with open area access points. Thanks to wireless signal measurement In Figure 1, the wireless network model designed in the application can be seen. When the structure of historical places is taken into consideration, there occurs the necessity of installing an efficient network with economical devices. In this topology, an economical mesh network has been formed over alternative routes. To enable fast data flow, devices supported with dipole antenna have been located in places called dead spot. In Figure 2, what was mostly considered was the determination of the correct point that will reach the devices to the extranet by locating the gateway of each device. The determination of the efficiency of the routes has been made by protecting the signal levels in efficient points. In figure 3, the channels that wireless access points use and signal levels taken from these channels can be seen. In the points where measurement has been performed, firstly, the improvements in points where signal levels are low have been taken into consideration. After determining the dipole angles in wireless mesh points, no problem has been observed in data flow in the channels.

Figure 3.The channels where wireless access points have been used

Table 1. Comparing data speed of wireless networks with the channels from which signals are sent [10] 4, Application The insufficiency in data security has affected the attitude towards wireless network negatively. A part from the studies related to the provision of safe data transfer, studies to increase wireless performance has been increasing [9]. In Table 1 given above, the channels over which signals are sent and data flow speed among these channels are observed [10]. When compared considering modulation coding plan, for the signals that will be sent over 40 Mhz, it is possible to say that data flow speed efficiency is enough. area [6]. In the literature review made, it has been observed that related to the installment of wireless networks needed in historical places comprehensible scientific studies haven’t been done. In the research done about the optimization methods of wireless networks without considering the places they are located on, at the installment stage, to reduce the cost, studies have been done related to the linear programming models and correct positioning. To increase the performance of wireless network, the optimization of predefined routes is necessary. Connection quality measurement has been performed metrically between the points and its effect on network performance has been observed with the media access control protocols developed, minimizing the poorness of data flow has been aimed. With the chain models

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Mixed discrete least squares meshless method for analysis of elasticity problems (in Persian)

by Mohammad Afshar

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A finite point method for adaptive three-dimensional compressible flow calculations

by Enrique Ortega

2009, International Journal for Numerical Methods in Fluids

The finite point method (FPM) is a meshless technique, which is based on both, a weighted least-squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a broader investigation into the capabilities of the FPM to deal with 3D applications concerning real compressible fluid flow problems. In the first part of this work, the upwind-biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting the meshless capabilities, an h-adaptive methodology for 2D and 3D compressible flow calculations is developed. This adaptive technique applies a solution-based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are encouraging. Several numerical examples are provided in order to illustrate the good performance of the numerical methods presented. resources and ever-challenging demands for practical and theoretical applications. Nowadays, there are two main types of numerical techniques for solving PDEs. On the one hand, there exist meshbased or conventional discretization methods; among them the classical finite differences (FD), finite volume (FV) and finite element (FE) methods are of singular interest. These techniques are mostly employed in practice due to their robustness, efficiency and high confidence gained through years of continuous use and enhancement. On the other hand, there exist meshless methods. Having their pros and cons, meshless methods offer an alternative to mesh-based techniques. Meshless methods are conceptually attractive; however, their practical implementations have not succeeded so far to prove their efficiency and this is a fact which can explain the comparatively little attention that has been devoted to these techniques. In spite of this, over the last 10 years, some difficulties that arose in conventional mesh-based methods when performing particular applications have brought meshless methods into the focus of attention.

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Node moving adaptive refinement strategy for planar elasticity problems using discrete least squares meshless method

by Mohammad Afshar

2011, Finite Elements in Analysis and Design

In this paper, an error indicator and adaptive refinement procedure in conjunction with the Discrete Least Squares Meshless (DLSM) method is presented for the effective and efficient analysis of planar elasticity problems. The DLSM method... more

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A Grid-Free Upwind Relaxation Scheme for Inviscid Compressible Flows

by Raghurama Rao, V. Suswaram

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Mixed Discrete Least Squares Meshless method for planar elasticity problems using regular and irregular nodal distributions

by Jafar Amani

2012, Engineering Analysis With Boundary Elements