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arxiv: 2604.23155 · v1 · submitted 2026-04-25 · 🧮 math.NA · cs.NA

Mesh-Intrinsic GFEM: High-Order Smoothness on C⁰ Unstructured Meshes

Rong Tian This is my paper

Pith reviewed 2026-05-08 07:45 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords generalized finite element methodpartition of unityC0 mesheshigh-order smoothnessstrong-form collocationunstructured meshesnumerical PDEsboundary absorption
0
0 comments X p. Extension

The pith

Local polynomial reconstructions on overlapping patches blended by partition of unity cancel derivative jumps exactly for polynomials on standard C0 meshes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how a mesh-intrinsic generalized finite element method reconstructs local polynomial fields from shared nodal values on overlapping patches and blends them with a partition of unity. This produces an interface coherence effect in which derivative jumps at element boundaries vanish exactly when the field is a polynomial and decay at the optimal rate O(h to the p plus one minus alpha) for smooth non-polynomial fields. A sympathetic reader would care because the construction supplies a polynomial-exact intrinsic derivative that supports pointwise strong-form evaluation of high-order PDEs while using only conventional C0 unstructured meshes and without introducing extra global degrees of freedom.

Core claim

The core analysis establishes a partition-of-zero smoothness-transfer mechanism driven by interface coherence: derivative jumps cancel exactly for polynomial reproduction and decay as O(h^{p+1-|α|}) for smooth nonpolynomial fields. On this basis a PoZ-consistent intrinsic derivative is defined that is polynomial-exact and approximation-order consistent, enabling pointwise strong-form evaluation of high-order PDEs on C0 meshes. For derivative-type or free boundary conditions the method embeds constraints into local patch reconstruction via a boundary absorption constrained weighted least-squares strategy that keeps the global system square and sparse.

What carries the argument

The partition-of-zero (PoZ) smoothness-transfer mechanism, which uses interface coherence arising from overlapping nodal patch reconstructions blended by a partition of unity to cancel derivative jumps at element interfaces.

If this is right

  • Machine-precision patch tests are obtained for all polynomials up to the reconstruction degree.
  • Derivative jump magnitudes decay at the rate O(h^{p+1-|α|}) for smooth non-polynomial fields.
  • The same trial space supports both weak-form Galerkin and strong-form collocation discretizations of high-order PDEs.
  • Boundary constraints are embedded directly into local patch solves without global over-determination or penalty-parameter tuning.
  • Robust accuracy is retained on highly distorted unstructured meshes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-order mesh generation may become unnecessary for many engineering problems governed by high-order PDEs.
  • Convergence behavior could be benchmarked directly against classical high-order finite-element codes to quantify savings in mesh generation and storage.
  • The smoothness-transfer idea may apply to other partition-of-unity frameworks such as meshfree or extended finite-element methods.

Load-bearing premise

Local polynomial reconstructions on overlapping nodal patches, when blended by a partition of unity, produce interface coherence sufficient for exact derivative-jump cancellation on polynomials and the stated decay rate on non-polynomials without additional global constraints or post-processing.

What would settle it

A sequence of refined meshes on which measured derivative jumps at element interfaces for a smooth non-polynomial field fail to decay at the predicted O(h^{p+1-|α|}) rate, or on which polynomial patch tests fail to reach machine precision.

Figures

Figures reproduced from arXiv: 2604.23155 by Rong Tian.

Figure 1
Figure 1. Figure 1: Illustration of the mesh-intrinsic GFEM framework in 1D: (a) Overlapping nodal patches and local view at source ↗
Figure 2
Figure 2. Figure 2: Patch node sets and stencils on an unstructured triangular mesh. (a) Interior patch with view at source ↗
Figure 3
Figure 3. Figure 3: Case 2: Asymptotic C p -continuity on triangular meshes with zero node perturbation for p = 3,4,5. The plots show the decay of pointwise L ∞ jump metrics Jm under uniform h-refinement, with dashed reference lines indicating the predicted rates O(h p+1−m). 8.3 High-Order Patch Tests This subsection checks polynomial consistency at the PDE level. Unlike the smoothness diagnostics in Section 8, these tests ve… view at source ↗
Figure 4
Figure 4. Figure 4: Mixed Dirichlet–Neumann elasticity benchmark: NC-QR, NC-BA, SD-QR, SD-BA, and WG view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of DOF layouts for MiGFEM WG (nodal DOFs), standard view at source ↗
Figure 6
Figure 6. Figure 6: DOF-based comparison of MiGFEM WG, standard view at source ↗
Figure 7
Figure 7. Figure 7: Conditioning against interior vector DOFs for MiGFEM (NC/CC/SD/WG), FEM P1, and FEM view at source ↗
Figure 8
Figure 8. Figure 8: Representative irregular 11×11 triangulations on (0,5) 2 obtained by perturbing interior nodes by fractions δ ∈ {0.3h,0.5h} of the local mesh size while keeping boundary nodes fixed. Results. Across perturbed meshes, all MiGFEM variants retain monotone convergence in L 2 , H 1 , and en￾ergy norms. NC/CC/SD and WG remain stable over the tested distortion levels, indicating robust CWLS reconstruction under t… view at source ↗
Figure 9
Figure 9. Figure 9: Elasticity benchmark on perturbed meshes with view at source ↗
Figure 10
Figure 10. Figure 10: Elasticity benchmark on perturbed meshes with increased distortion ( view at source ↗
Figure 11
Figure 11. Figure 11: WG post-processing comparison of Leibniz and intrinsic derivatives for view at source ↗
Figure 12
Figure 12. Figure 12: Representative 11 × 11 meshes for the biharmonic benchmark with perturbation levels δ = 0.0h,0.5h,0.8h,1.0h. 8.5.2 Approximation-space diagnostic: exact nodal injection Objective. To isolate the approximation-space capability from algebraic solve and boundary-enforcement effects. Setup. We inject exact nodal values of the analytical solution. NC/CC/SD therefore share the same recon￾structed field, and the… view at source ↗
Figure 13
Figure 13. Figure 13: Nodal-injection diagnostic for the biharmonic benchmark at view at source ↗
Figure 14
Figure 14. Figure 14: Nodal-injection diagnostic for the biharmonic benchmark at view at source ↗
Figure 15
Figure 15. Figure 15: Full-solve convergence for biharmonic NC/CC/SD at view at source ↗
Figure 16
Figure 16. Figure 16: Full-solve convergence for biharmonic NC/CC/SD at view at source ↗
Figure 17
Figure 17. Figure 17: NC biharmonic comparison for p = 4: CWLS-Penalty, CWLS-BA, and RBF-Penalty. Dashed reference line: O(h 2 ). 35 view at source ↗
Figure 18
Figure 18. Figure 18: NC biharmonic comparison for p = 6: CWLS-Penalty, CWLS-BA, and RBF-Penalty. Dashed reference line: O(h 4 ) view at source ↗
Figure 19
Figure 19. Figure 19: Global conditioning and total-runtime comparison for view at source ↗
Figure 20
Figure 20. Figure 20: Global conditioning and total-runtime comparison for view at source ↗
read the original abstract

High-order partial differential equations (PDEs) require derivative regularity that standard $C^0$ finite element infrastructures do not directly provide on unstructured meshes. We propose a mesh-intrinsic generalized finite element method (MiGFEM) that reconstructs local polynomial fields on overlapping nodal patches from shared nodal unknowns and blends them by a partition of unity, without introducing extra global degrees of freedom. The core analysis establishes a partition-of-zero (PoZ) smoothness-transfer mechanism driven by interface coherence: derivative jumps cancel exactly for polynomial reproduction and decay as $O(h^{p+1-|\alpha|}) $for smooth nonpolynomial fields. On this basis, we define a PoZ-consistent intrinsic derivative that is polynomial-exact and approximation-order consistent, enabling pointwise strong-form evaluation of high-order PDEs on $C^0$ meshes. For derivative-type/free boundary conditions in strong-form collocation,we introduce a boundary absorption constrained weighted least-squares strategy (BA-CWLS), which embeds boundary constraints into local patch reconstruction. This avoids globally overdetermined boundary augmentation and penalty tuning, while preserving a square sparse global system. Numerical experiments show machine-precision patch tests,jump-decay rates consistent with theory, and robust performance on highly distorted meshes. The same mesh-intrinsic trial space supports both weak-form Galerkin and strong-form collocation discretizations, providing a unified high-order route on standard $C^0$ mesh infrastructures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims to introduce a mesh-intrinsic generalized finite element method (MiGFEM) for high-order smoothness on C^0 unstructured meshes. Local polynomial fields are reconstructed on overlapping nodal patches from shared nodal unknowns and blended using a partition of unity. A partition-of-zero (PoZ) smoothness-transfer mechanism is established, driven by interface coherence, leading to exact derivative jump cancellation for polynomials and O(h^{p+1-|α|}) decay for smooth nonpolynomial fields. A PoZ-consistent intrinsic derivative is defined for strong-form PDE evaluation, and a boundary absorption constrained weighted least-squares (BA-CWLS) strategy is proposed for boundary conditions. Numerical experiments demonstrate machine-precision patch tests, consistent jump-decay rates, and robustness on distorted meshes, supporting both Galerkin and collocation approaches.

Significance. If the PoZ mechanism and interface coherence hold as described, the work is significant for enabling high-order PDE discretizations on standard C^0 meshes without extra global degrees of freedom or special meshing. The unified trial space for weak-form Galerkin and strong-form collocation, combined with the reported machine-precision patch tests and matching theoretical decay rates, represents a practical strength. The mesh-intrinsic construction and BA-CWLS boundary handling avoid common pitfalls in generalized FEM extensions.

minor comments (2)
  1. The numerical results section mentions machine-precision patch tests and jump-decay rates consistent with theory but supplies no explicit error tables, figures, or quantitative data; including these would allow independent verification of the central claims.
  2. Notation for the PoZ-consistent intrinsic derivative and the precise definition of interface coherence would benefit from a dedicated equation or definition box in the analysis section for improved clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and for recommending minor revision. The referee's description accurately reflects the core elements of MiGFEM, including the partition-of-zero smoothness-transfer mechanism, interface coherence, polynomial-exact intrinsic derivatives, BA-CWLS boundary treatment, and the supporting numerical results. We appreciate the recognition of the method's significance for high-order discretizations on standard C^0 unstructured meshes without additional global degrees of freedom.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs local polynomial reconstructions on overlapping nodal patches from shared nodal unknowns, then blends via partition of unity. The core analysis derives the PoZ smoothness-transfer property (exact jump cancellation on polynomials, O(h^{p+1-|α|}) decay otherwise) directly from this construction and interface coherence induced by the shared unknowns. The PoZ-consistent intrinsic derivative is then defined on that basis. Numerical patch tests at machine precision and observed jump rates serve as independent external checks rather than tautological confirmation. No quoted step reduces a claimed prediction or uniqueness result to a self-definition, a fitted parameter, or a self-citation chain; the derivation remains self-contained against the stated assumptions and supporting experiments.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 3 invented entities

The central claim rests on the newly introduced PoZ mechanism and BA-CWLS strategy, which are defined inside the paper; standard partition-of-unity and polynomial-reproduction properties are invoked but the smoothness-transfer analysis itself is not anchored to external benchmarks or machine-checked proofs in the abstract.

axioms (2)
  • standard math Partition of unity property holds for the blending functions on overlapping patches
    Invoked to combine local fields without introducing extra global degrees of freedom.
  • domain assumption Local reconstructions exactly reproduce polynomials up to degree p
    Required for the exact cancellation of derivative jumps under the PoZ mechanism.
invented entities (3)
  • partition-of-zero (PoZ) smoothness-transfer mechanism no independent evidence
    purpose: To guarantee exact cancellation of derivative jumps for polynomials and controlled decay for smooth fields across patch interfaces
    New analytic construct introduced to characterize the smoothness transfer on C0 meshes.
  • PoZ-consistent intrinsic derivative no independent evidence
    purpose: To provide a pointwise, polynomial-exact derivative operator for strong-form evaluation of high-order PDEs
    Defined directly from the PoZ property and local reconstructions.
  • boundary absorption constrained weighted least-squares strategy (BA-CWLS) no independent evidence
    purpose: To embed derivative-type or free boundary conditions into local patch reconstructions without global over-determination or penalty parameters
    New boundary-handling technique presented as part of the method.

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Reference graph

Works this paper leans on

83 extracted references · 7 canonical work pages

  1. [1]

    Zienkiewicz, R.L

    O.C. Zienkiewicz, R.L. Taylor. The Finite Element Method: Its Basis and Fundamentals, 7th ed., Butterworth-Heinemann, 2013

    2013

  2. [2]

    K.J. Bathe. Finite Element Procedures, 2nd ed., Prentice Hall, 2014

    2014

  3. [3]

    T.J.R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Prentice-Hall International, Englewood Cliffs, New Jersey, 1987

    1987

  4. [4]

    Brenner, L.R

    S.C. Brenner, L.R. Scott. The Mathematical Theory of Finite Element Methods, Springer, 2008

    2008

  5. [5]

    Ern, J.-L

    A. Ern, J.-L. Guermond. Theory and Practice of Finite Elements, Springer, 2004

    2004

  6. [6]

    Timoshenko, S

    S. Timoshenko, S. Woinowsky-Krieger. Theory of Plates and Shells, 2nd ed., McGraw–Hill, New York, 1959

    1959

  7. [7]

    J.N. Reddy. Theory and Analysis of Elastic Plates and Shells, CRC Press, Taylor and Francis, 2007

    2007

  8. [8]

    A.C. Eringen. Nonlocal Continuum Field Theories, Springer, New York, 2001

    2001

  9. [9]

    R.D. Mindlin. Second gradient of strain and surface-tension in linear elasticity, International Journal of Solids and Structures 1 (1965) 417–438

    1965

  10. [10]

    Aifantis

    E.C. Aifantis. On the role of gradients in the localization of deformation and fracture, International Journal of Engineering Science 30 (1992) 1279–1299

    1992

  11. [11]

    G. A. Maugin. Non-Classical Continuum Mechanics: A Dictionary, Springer, 2017

    2017

  12. [12]

    Miehe, F

    C. Miehe, F. Welschinger, M. Hofacker. Thermodynamically consistent phase-field models of frac- ture: Variational principles and multi-field FE implementations, International Journal for Numerical Methods in Engineering 83 (2010) 1273–1311

    2010

  13. [13]

    Cahn, J.E

    J.W. Cahn, J.E. Hilliard. Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics 28 (1958) 258–267. 40

    1958

  14. [14]

    Liangzhe Zhang, M. R. Tonks, D. Gaston, J.W. Peterson, D. Andrs, P.C. Millett, B.S. Biner. A quan- titative comparison betweenC 0 andC 1 elements for solving the Cahn–Hilliard equation, Journal of Computational Physics 236 (2013) 74–80

    2013

  15. [15]

    H. Lamb. Hydrodynamics, 6th ed., Cambridge University Press, Cambridge, 1994

    1994

  16. [16]

    Clough, J.L

    R.W. Clough, J.L. Tocher. Finite Element Stiffness Matrices for Analysis of Plates in Bending, Pro- ceedings of the Conference on Matrix Methods in Structural Mechanics, Wright-Patterson Air Force Base, 1965, pp. 515–545

    1965

  17. [17]

    Argyris, I

    J.H. Argyris, I. Fried, D.W. Scharpf. The TUBA Family of Plate Elements for the Matrix Displacement Method, Aeronautical Journal of the Royal Aeronautical Society, 72(692), 1968: 701–709

    1968

  18. [18]

    Bogner, R.L

    F.K. Bogner, R.L. Fox, L.A. Schmit. The Generation of Interelement Compatible Stiffness and Mass Matrices by the Use of Interpolation Formulas, Wright-Patterson Air Force Base, Ohio, 1965, pp. 395–443

    1965

  19. [19]

    M. Zlámal. The Finite Element Method in Domains with Curved Boundaries, International Journal for Numerical Methods in Engineering 5(3) (1973) 367–373

    1973

  20. [20]

    Ciarlet, P.A

    P.G. Ciarlet, P.A. Raviart. Interpolation Theory over Curved Elements, with Applications to Finite Element Methods, Computer Methods in Applied Mechanics and Engineering 1(2) (1972) 217–249

    1972

  21. [21]

    P.G. Ciarlet. The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 2012

    2012

  22. [22]

    DOI: 10.6052/0459-1879-18-188

    Rong Tian, A GFEM withC 1-continuity, Chinese Journal of Theoretical and Applied Mechanics 51 (1) (2019) 263–277. DOI: 10.6052/0459-1879-18-188

    work page doi:10.6052/0459-1879-18-188 2019

  23. [23]

    Hughes, J.A

    T.J.R. Hughes, J.A. Cottrell, Y . Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering 194 (39–

  24. [24]

    Kiendl, R

    J. Kiendl, R. Schmidt, R. Wüchner, K.-U. Bletzinger. Isogeometric shell analysis with Kirchhoff–Love elements, Computer Methods in Applied Mechanics and Engineering 198 (49–52) (2009) 3902–3914

    2009

  25. [25]

    G.R. Liu. Meshfree Methods: Moving Beyond the Finite Element Method, 2nd ed., CRC Press, Boca Raton, 2009

    2009

  26. [26]

    Majeed, F

    M. Majeed, F. Cirak. Isogeometric analysis using manifold-based smooth basis functions, Computer Methods in Applied Mechanics and Engineering 316 (2017) 758–781

    2017

  27. [27]

    Zhang, F

    Q. Zhang, F. Cirak, Manifold-based isogeometric analysis basis functions with prescribed sharp fea- tures, Computer Methods in Applied Mechanics and Engineering 359 (2020) 112659

    2020

  28. [28]

    J.-S. Chen, M. Hillman, S.-W. Chi, Meshfree methods: Progress made after 20 years, Computational Mechanics 59 (6) (2017) 733–755. DOI: 10.1007/s00466-017-1414-1

    work page doi:10.1007/s00466-017-1414-1 2017

  29. [29]

    Belytschko, Y .Y

    T. Belytschko, Y .Y . Lu, L. Gu, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37(2) (1994) 229–256

    1994

  30. [30]

    W.K. Liu, S. Jun, Y .F. Zhang, Reproducing kernel particle methods, Computational Mechanics 20 (1995) 108–116. DOI: 10.1007/BF00350275. 41

    work page doi:10.1007/bf00350275 1995

  31. [31]

    Kansa, Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I

    E.J. Kansa, Multiquadrics—a scattered data approximation scheme with applications to computational fluid-dynamics—I. Surface approximations and partial derivative estimates, Computers & Mathemat- ics with Applications 19 (8–9) (1990) 127–145. DOI: 10.1016/0898-1221(90)90109-6

    work page doi:10.1016/0898-1221(90)90109-6 1990

  32. [32]

    Larsson, E

    E. Larsson, E. Lehto, A. Heryudono, B. Fornberg, Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions, SIAM Journal on Scientific Comput- ing 35 (4) (2013) A2096–A2119

    2013

  33. [33]

    Fornberg, N

    B. Fornberg, N. Flyer, Solving PDEs with radial basis functions, Acta Numerica 24 (2015) 215–258

    2015

  34. [34]

    Flyer, G.B

    N. Flyer, G.B. Wright, B. Fornberg, Radial basis function-generated finite differences: A mesh-free method for computational geosciences, in: Handbook of Geomathematics, Springer, 2015, pp. 1–30

    2015

  35. [35]

    C. Shu, H. Ding, K.S. Yeo, Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering 192 (7–8) (2003) 941–954

    2003

  36. [36]

    To, WingKam Liu, Conforming local meshfree method, International Journal for Numerical Methods in Engineering 86 (2011) 335–357

    Rong Tian, Albert C. To, WingKam Liu, Conforming local meshfree method, International Journal for Numerical Methods in Engineering 86 (2011) 335–357

    2011

  37. [37]

    G. H. Shi, Manifold method of material analysis, in: Transactions of the 9th Army Conference on Applied Mathematics and Computing, No. 92-1, U.S. Army Research Office, Minneapolis, Minnesota, (1991) 57–76

    1991

  38. [38]

    G. H. Shi, Modeling rock joints and blocks by manifold method, in: Proceedings of the 33rd U.S. Symposium on Rock Mechanics, American Rock Mechanics Association, USRMS, Santa Fe, New Mexico, 1992, pp. 639–648

    1992

  39. [39]

    Babuška, J.M

    I. Babuška, J.M. Melenk. The partition of unity method, International Journal for Numerical Methods in Engineering 40(4) (1997) 727–758

    1997

  40. [40]

    Melenk, I

    J.M. Melenk, I. Babuška, The partition of unity finite element method: Basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 (1996) 289–314

    1996

  41. [41]

    Strouboulis, I

    T. Strouboulis, I. Babuška, K. Copps, The design and analysis of the Generalized Finite Element Method, Computer Methods in Applied Mechanics and Engineering 181 (2000) 43–69

    2000

  42. [42]

    C. A. Duarte, I. Babuška, J. T. Oden, Generalized finite element methods for three-dimensional struc- tural mechanics problems, Computers & Structures 77 (2000) 215–232

    2000

  43. [43]

    Belytschko, T

    T. Belytschko, T. Black, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (1999) 601–620

    1999

  44. [44]

    N. Moës, J. Dolbow, T. Belytschko, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46(1) (1999) 131–150

    1999

  45. [45]

    Strouboulis, K

    T. Strouboulis, K. Copps, I. Babuška, The generalized finite element method, Computer Methods in Applied Mechanics and Engineering 190 (2001) 4081–4193

    2001

  46. [46]

    Rong Tian, Genki Yagawa, Haruo Terasaka, Linear dependence problems of partition of unity-based generalized FEMs, Computer Methods in Applied Mechanics and Engineering 195 (37–40) (2006) 4768–4782. 42

    2006

  47. [47]

    Rong Tian, A PU-based 4-node quadratic tetrahedron and linear dependence elimination in three di- mensions, International Journal of Computational Methods 3 (2006) 545–562

    2006

  48. [48]

    Rong Tian, Genki Yagawa, Generalized node and high-performance elements, International Journal for Numerical Methods in Engineering 64 (2005) 2039–2071

    2005

  49. [49]

    Rong Tian, Hiroshi Matsubara, Genki Yagawa, Advanced 4-node tetrahedrons, International Journal for Numerical Methods in Engineering 68 (2006) 1209–1231

    2006

  50. [50]

    Rong Tian, Extra-DOF-free and linearly independent enrichments in GFEM, Computer Methods in Applied Mechanics and Engineering 266 (2013) 1–22

    2013

  51. [51]

    Holzer, The mixed-cell-complex partition-of-unity method, Computer Meth- ods in Applied Mechanics and Engineering 198 (13–14) (2009) 1235–1248

    Carsten Riker, Stefan M. Holzer, The mixed-cell-complex partition-of-unity method, Computer Meth- ods in Applied Mechanics and Engineering 198 (13–14) (2009) 1235–1248

    2009

  52. [52]

    X.M. An, L.X. Li, G.W. Ma, H.H. Zhang, Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes, Computer Methods in Applied Mechanics and Engineering 200 (2011) 665–674

    2011

  53. [53]

    X.M. An, Z.Y . Zhao, H.H. Zhang, L.X. Li, Investigation of linear dependence problem of three- dimensional partition of unity-based finite element methods, Computer Methods in Applied Mechanics and Engineering 233–236 (2012) 137–151

    2012

  54. [54]

    Babuška, U

    I. Babuška, U. Banerjee, Stable Generalized Finite Element Method (SGFEM), Computer Methods in Applied Mechanics and Engineering 201–204 (2012) 91–111

    2012

  55. [55]

    Sillem, A

    A. Sillem, A. Simone, L.J. Sluys, The Orthonormalized Generalized Finite Element Method–OGFEM: Efficient and stable reduction of approximation errors through multiple orthonormalized enriched basis functions, Computer Methods in Applied Mechanics and Engineering 287 (2015) 112–149

    2015

  56. [56]

    Qinghui Zhang, Ivo Babuška, Uday Banerjee, Robustness in stable generalized finite element meth- ods (SGFEM) applied to Poisson problems with crack singularities, Computer Methods in Applied Mechanics and Engineering 311 (2016) 476–502

    2016

  57. [57]

    Qinghui Zhang, Ivo Babuška, A stable generalized finite element method (SGFEM) of degree two for interface problems, Computer Methods in Applied Mechanics and Engineering 363 (2020) 112889

    2020

  58. [58]

    Lancaster, K

    C. Lancaster, K. Salkauskas, Surfaces generated by moving least squares methods, Mathematics of Computation 37 (155) (1981) 141–158

    1981

  59. [59]

    Levin, The approximation power of moving least-squares, Mathematics of Computation 67 (224) (1998) 1517–1531

    D. Levin, The approximation power of moving least-squares, Mathematics of Computation 67 (224) (1998) 1517–1531

    1998

  60. [60]

    Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2004

    H. Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2004

    2004

  61. [61]

    Rong Tian, LongFei Wen, Improved XFEM — An extra-DOF-free, well-conditioned, and interpolating XFEM, Computer Methods in Applied Mechanics and Engineering 285 (2015) 639–658

    2015

  62. [62]

    LongFei Wen, Rong Tian, Improved XFEM: Accurate and robust dynamic crack growth simulation, Computer Methods in Applied Mechanics and Engineering 308 (2016) 256–285

    2016

  63. [63]

    Rong Tian, LongFei Wen, LiXiang Wang, Three-dimensional improved XFEM (IXFEM) for static crack problems, Computer Methods in Applied Mechanics and Engineering 343 (2019) 339–367. 43

    2019

  64. [64]

    DOI: 10.1016/j.cma.2020.113659

    Guizhong Xiao, LongFei Wen, Rong Tian, Arbitrary 3D crack propagation with Improved XFEM: Accurate and efficient crack geometries, Computer Methods in Applied Mechanics and Engineering 377 (2021) 113659. DOI: 10.1016/j.cma.2020.113659

    work page doi:10.1016/j.cma.2020.113659 2021

  65. [65]

    DOI: 10.1016/j.cma.2022.115844

    Guizhong Xiao, LongFei Wen, Rong Tian, Dingguo Zhang, Improved XFEM (IXFEM): 3D dynamic crack propagation under impact loading, Computer Methods in Applied Mechanics and Engineering 405 (2023) 115844. DOI: 10.1016/j.cma.2022.115844

    work page doi:10.1016/j.cma.2022.115844 2023

  66. [66]

    JinWei Ma, QingLin Duan, Rong Tian, A generalized finite element method without extra degrees of freedom for large deformation analysis of three-dimensional elastic and elastoplastic solids, Computer Methods in Applied Mechanics and Engineering 392 (2022) 114639

    2022

  67. [67]

    LongFei Wen, Rong Tian, LiXiang Wang, Chun Feng, Improved XFEM for multiple crack analy- sis: Accurate and efficient implementations for stress intensity factors, Computer Methods in Applied Mechanics and Engineering 411 (2023) 116045

    2023

  68. [68]

    LiXiang Wang, LongFei Wen, Rong Tian, Chun Feng, Improved XFEM (IXFEM): Arbitrary multiple crack initiation, propagation and interaction analysis, Computer Methods in Applied Mechanics and Engineering 421 (2024) 116791

    2024

  69. [69]

    Hansbo, A

    P. Hansbo, A. Hansbo, An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems, Computer Methods in Applied Mechanics and Engineering 191 (2002) 5537–5552

    2002

  70. [70]

    Perrone, R

    N. Perrone, R. Kao, A general finite difference method for arbitrary meshes, Computers & Structures 5 (1) (1975) 45–58

    1975

  71. [71]

    Liszka, J

    T. Liszka, J. Orkisz, The finite difference method at arbitrary irregular grids and its application in applied mechanics, Computers & Structures 11 (1–2) (1980) 83–95

    1980

  72. [72]

    Benito, F

    J.J. Benito, F. Ureña, L. Gavete, Influence of several factors in the generalized finite difference method, Applied Mathematical Modelling 25 (12) (2001) 1039–1053

    2001

  73. [73]

    J. Slak, G. Kosec, Medusa: A C++ Library for Solving PDEs Using Strong Form Mesh-free Methods, ACM Transactions on Mathematical Software 47 (3) (2021), Article 28, 25 pages. DOI: 10.1145/3450966

    work page doi:10.1145/3450966 2021

  74. [74]

    Tinoco-Guerrero, F.J

    G. Tinoco-Guerrero, F.J. Domínguez-Mota, J.A. Guzmán-Torres, G. Pedraza-Jiménez, J.G. Tinoco- Ruiz, mGFD: A meshless generalized finite difference method, Computers & Mathematics with Ap- plications 195 (2025) 396–418

    2025

  75. [75]

    Fornberg, E

    B. Fornberg, E. Lehto, C. Powell, Stable calculation of Gaussian-based RBF-FD stencils, Computers & Mathematics with Applications 65 (4) (2013) 627–637

    2013

  76. [76]

    Flyer, B

    N. Flyer, B. Fornberg, V . Bayona, G.A. Barnett, On the role of polynomials in RBF-FD approxima- tions: I. Interpolation and accuracy, Journal of Computational Physics 321 (2016) 21–38

    2016

  77. [77]

    Bayona, N

    V . Bayona, N. Flyer, B. Fornberg, G.A. Barnett, On the role of polynomials in RBF-FD approxima- tions: II. Numerical solution of elliptic PDEs, Journal of Computational Physics 332 (2017) 257–273

    2017

  78. [78]

    Kuhn, A.W

    H.W. Kuhn, A.W. Tucker, Nonlinear Programming, in: Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, 1951, pp. 481– 492. 44

    1951

  79. [79]

    Nocedal, S.J

    J. Nocedal, S.J. Wright, Numerical Optimization, 2nd ed., Springer, New York, 2006

    2006

  80. [80]

    Christiansen, J

    S.H. Christiansen, J. Hu, K. Hu, Nodal finite element de Rham complexes, Numerische Mathematik 139 (2) (2018) 411–446

    2018

Showing first 80 references.