Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

4. Galerkin Methods

verfasst von : Marcus Olavi Rüter

Erschienen in: Error Estimates for Advanced Galerkin Methods

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

Having derived in the previous chapter various boundary value problems, including the finite and linearized hyperelasticity problems for both compressible and (nearly) incompressible materials, a reasonable question is how these problems can be solved. For most cases in engineering practice, the problems, including their geometry, are too complex for the feasible derivation of an exact analytical solution even though such a solution exists. We are therefore forced to employ numerical methods to obtain, at least, approximate solutions to the boundary value problems stated in the previous chapter.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Boundary Value Problems
Nächstes Kapitel Numerical Integration
Fußnoten
1
In the literature, the Galerkin method is also referred to as the Rayleigh-Ritz-Galerkin or the Ritz-Galerkin method to honor the works of John William Strutt, 3rd Baron Rayleigh (1842–1919) and Walther Ritz (1878–1909) that serve as a basis for the Galerkin method.
 
2
The reader is reminded that both \({\mathcal T}\) and \({\mathcal V}\) were already introduced in Sect. 3.​1.​3.
 
3
The error measure \(E_2\) is an artificial error measure. Therefore, the factor 1/2 is optional and does not affect the value of \(\varvec{a}\). We include this factor because it provides consistency with quadratic functionals of a physical nature, such as virtually all energy functionals.
 
Literatur
Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, New York (2000)
Aluru, N.R.: A point collocation method based on reproducing kernel approximations. Int. J. Numer. Meth. Engng. 47, 1083–1121 (2000)MATHCrossRef
Arnold, D.N., Boffi, D., Falk, R.S.: Quadrilateral \({H}(\rm div)\) finite elements. SIAM J. Numer. Anal. 42, 2429–2451 (2005)MathSciNetMATHCrossRef
Arnold, D.N., Brezzi, F., Douglas, J.: PEERS: A new mixed finite element for plane elasticity. Japan J. Appl. Math. 1, 347–367 (1984)MathSciNetMATHCrossRef
Arroyo, M., Ortiz, M.: Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int. J. Numer. Meth. Engng. 65, 2167–2202 (2006)MathSciNetMATHCrossRef
Auricchio, F., Beirão da Veiga, L., Brezzi, F., Lovadina, C.: Mixed finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 1, 2, pp. 149–201. John Wiley & Sons, Chichester (2017)
Babuška, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: A unified approach. Acta Numer. 1–125 (2003)
Backus, G., Gilbert, F.: The resolving power of gross earth data. Geophys. J. R. astr. Soc. 16, 169–205 (1968)MATHCrossRef
Bathe, K.-J.: Finite Element Procedures, 2nd edn. K.-J Bathe, Watertown (2014)MATH
Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Engng. 45, 601–620 (1999)MATHCrossRef
Belytschko, T., Gracie, R.: On XFEM applications to dislocations and interfaces. Int. J. Plast. 23, 1721–1738 (2007)MATHCrossRef
Belytschko, T., Krongauz, Y., Dolbow, J., Gerlach, C.: On the completeness of meshfree particle methods. Int. J. Numer. Meth. Engng. 43, 785–819 (1998)MathSciNetMATHCrossRef
Belytschko, T., Krongauz, Y., Fleming, M., Organ, D., Liu, W.K.: Smoothing and accelerated computations in the element free Galerkin method. J. Comput. Appl. Math. 74, 111–126 (1996a)MathSciNetMATHCrossRef
Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: An overview and recent developments. Comput. Methods Appl. Mech. Engrg. 139, 3–47 (1996b)MATHCrossRef
Belytschko, T., Liu, W.K., Moran, B., Elkhodary, K.I.: Nonlinear Finite Elements for Continua and Structures, 2nd edn. John Wiley & Sons, Chichester (2014)MATH
Bertsekas, D.P.: Constrained Optimization and Lagrange Multiplier Methods. Athena Scientific, Belmont (1996)MATH
Bochev, P.B., Gunzburger, M.D.: Least-Squares Finite Element Methods. Springer, New York (2009)MATH
Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)MATHCrossRef
Boiveau, T., Burman, E.: A penalty-free Nitsche method for the weak imposition of boundary conditions in compressible and incompressible elasticity. IMA J. Numer. Anal. 36, 770–795 (2016)MathSciNetMATHCrossRef
Bonet, J., Kulasegaram, S.: Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations. Int. J. Numer. Meth. Engng. 47, 1189–1214 (2000)MATHCrossRef
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)MATHCrossRef
Bracewell, R.N.: The Fourier Transform and its Applications, 3rd edn. McGraw-Hill, New York (1999)MATH
Braess, D.: Finite Elements—Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, Cambridge (2007)
Brenner, S.C., Carstensen, C.: Finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 1, 2nd edn., pp. 101–147. John Wiley & Sons, Chichester (2017)
Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)MATHCrossRef
Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8, 129–151 (1974)MathSciNetMATHCrossRef
Brezzi, F., Douglas Jr., J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47, 217–235 (1985)MathSciNetMATHCrossRef
Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2003)MATHCrossRef
Burman, E.: A penalty free nonsymmetric Nitsche-type method for the weak imposition of boundary conditions. SIAM J. Numer. Anal. 50, 1959–1981 (2012)MathSciNetMATHCrossRef
Chen, J.S., Belytschko, T.: Meshless and meshfree methods. In: Engquist, B. (ed.) Encyclopedia of Applied and Computational Mathematics, pp. 886–894. Springer, Heidelberg (2015)CrossRef
Chen, J.S., Han, W., You, Y., Meng, X.: A reproducing kernel method with nodal interpolation property. Int. J. Numer. Meth. Engng. 56, 935–960 (2003)MathSciNetMATHCrossRef
Chen, J.S., Liu, W.K., Hillman, M.C., Chi, S.W., Lian, Y., Bessa, M.A.: Reproducing kernel particle method for solving partial differential equations. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 2, 2nd edn., pp. 691–734. John Wiley & Sons, Chichester (2017)
Chen, J.S., Pan, C., Roque, C.M.O.L., Wang, H.P.: A Lagrangian reproducing kernel particle method for metal forming analysis. Comput. Mech. 22, 289–307 (1998)MATHCrossRef
Chen, J.S., Pan, C., Wu, C.T., Liu, W.K.: Reproducing Kernel Particle Methods for large deformation analysis of non-linear structures. Comput. Methods Appl. Mech. Engrg. 139, 195–227 (1996)MathSciNetMATHCrossRef
Chen, J.S., Wang, H.P.: New boundary condition treatments in meshfree computation of contact problems. Comput. Methods Appl. Mech. Engrg. 187, 441–468 (2000)MathSciNetMATHCrossRef
Chi, S.W., Chen, J.S., Hu, H.Y., Yang, J.P.: A gradient reproducing kernel collocation method for boundary value problems. Int. J. Numer. Meth. Engng. 93, 1381–1402 (2013)MathSciNetMATHCrossRef
Christ, N.H., Friedberg, R., Lee, T.D.: Weights of links and plaquettes in a random lattice. Nucl. Phys. B 210, 337–346 (1982)MathSciNetCrossRef
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978)MATH
Courant, R.: Variational methods for the solution of problems of equilibrium and vibrations. Bull. Amer. Math. Soc. 49, 1–23 (1943)MathSciNetMATHCrossRef
de Borst, R., Crisfield, M.A., Remmers, J.J.C., Verhoosel, C.V.: Non-linear Finite Element Analysis of Solids and Structures, 2nd edn. John Wiley & Sons, Chichester (2012)MATHCrossRef
Duarte, C.A., Oden, J.T.: An \(h\)-\(p\) adaptive method using clouds. Comput. Methods Appl. Mech. Engrg. 139, 237–262 (1996a)MathSciNetMATHCrossRef
Duarte, C.A., Oden, J.T.: \(H\)-\(p\) clouds—an \(h\)-\(p\) meshless method. Numer. Methods Partial Differential Eq. 12, 673–705 (1996b)MathSciNetMATHCrossRef
Duchon, J.: Splines minimizing rotation-invariant semi-norms in Sobolev spaces. Constructive Theory of Functions of Several Variables. Lecture Notes in Mathematics, vol. 571, pp. 85–100. Springer, Berlin (1976)CrossRef
Düster, A., Rank, E., Szabó, B.: The \(p\)-version of the finite element and finite cell methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, 2 edn., pp. 137–171. John Wiley & Sons, Chichester (2017)
Fasshauer, G.E.: Approximate moving least-squares approximation with compactly supported radial weights. In: Griebel, M., Schweitzer, M.A. (eds.) Meshfree Methods for Partial Differential Equations, pp. 105–116. Springer, Berlin (2003)CrossRef
Fleming, M., Chu, Y.A., Moran, B., Belytschko, T.: Enriched element-free Galerkin methods for crack tip fields. Int. J. Numer. Meth. Engng. 40, 1483–1504 (1997)MathSciNetCrossRef
Fletcher, R.: Practical Methods of Optimization, 2nd edn. John Wiley & Sons, Chichester (2000)MATHCrossRef
Fries, T.-P.: A corrected XFEM approximation without problems in blending elements. Int. J. Numer. Meth. Engng. 75, 503–532 (2008)MathSciNetMATHCrossRef
Fries, T.-P., Belytschko, T.: The intrinsic XFEM: a method for arbitrary discontinuities without additional unknowns. Int. J. Numer. Meth. Engng. 68, 1358–1385 (2006)MATHCrossRef
Fries, T.-P., Byfut, A., Alizada, A., Cheng, K.W., Schröder, A.: Hanging nodes and XFEM. Int. J. Numer. Meth. Engng. 86, 404–430 (2011)MathSciNetMATHCrossRef
Galerkin, B.G.: Series solutions of some cases of equilibrium of elastic beams and plates (in Russian). Vestn. Inshenernov. 1, 897–908 (1915)
Gaul, L.: From Newton’s principia via Lord Rayleigh’s theory of sound to finite elements. In: Stein, E. (ed.) The History of Theoretical, Material and Computational Mechanics—Mathematics meets Mechanics and Engineering, pp. 385–398. Springer, Berlin (2014)CrossRef
Gerasimov, T., Rüter, M., Stein, E.: An explicit residual-type error estimator for \(\mathbb{Q}_1\)-quadrilateral extended finite element method in two-dimensional linear elastic fracture mechanics. Int. J. Numer. Meth. Engng. 90, 1118–1155 (2012)MATHCrossRef
Gill, P.E., Murray, W., Wright, M.H.: Practical Optimization. Academic Press, London (1981)MATH
Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. astr. Soc. 181, 375–389 (1977)MATHCrossRef
Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations. Springer, Berlin (1986)MATHCrossRef
Griebel, M., Schweitzer, M.A.: A particle-partition of unity method, Part V: Boundary conditions. In: Hildebrandt, S., Karcher, H. (eds.) Geometric Analysis and Nonlinear Partial Differential Equations, pp. 519–542. Springer, Berlin (2003)CrossRef
Griebel, M., Schweitzer, M.A. (eds.): Meshfree Methods for Partial Differential Equations I–VIII. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2003–2017)
Günther, F.C., Liu, W.K.: Implementation of boundary conditions for meshless methods. Comput. Methods Appl. Mech. Engrg. 163, 205–230 (1998)MathSciNetMATHCrossRef
Han, W., Meng, X.: Error analysis of reproducing kernel particle method. Comput. Methods Appl. Mech. Engrg. 190, 6157–6181 (2001)MathSciNetMATHCrossRef
Hansbo, A., Hansbo, P.: A finite element method for the simulation of strong and weak discontinuities in solid mechanics. Comput. Methods Appl. Mech. Engrg. 193, 3523–3540 (2004)MathSciNetMATHCrossRef
Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 76, 1905–1915 (1971)CrossRef
Hattori, G., Rojas-Díaz, R., Sáez, A., Sukumar, N., García-Sánchez, F.: New anisotropic crack-tip enrichment functions for the extended finite element method. Comput. Mech. 50, 591–601 (2012)MathSciNetMATHCrossRef
Hegen, D.: Element-free Galerkin methods in combination with finite element approaches. Comput. Methods Appl. Mech. Engrg. 135, 143–166 (1996)MATHCrossRef
Holl, M., Loehnert, S., Wriggers, P.: An adaptive multiscale method for crack propagation and crack coalescence. Int. J. Numer. Meth. Engng. 93, 23–51 (2013)MathSciNetMATHCrossRef
Hu, H.Y., Chen, J.S., Hu, W.: Weighted radial basis collocation method for boundary value problems. Int. J. Numer. Meth. Engng. 69, 2736–2757 (2007)MathSciNetMATHCrossRef
Hu, H.Y., Chen, J.S., Hu, W.: Error analysis of collocation method based on reproducing kernel approximation. Numer. Methods Partial Differential Eq. 27, 554–580 (2011)MathSciNetMATHCrossRef
Huerta, A., Belytschko, T., Fernández-Méndez, S., Rabczuk, T., Zhuang, X., Arroyo, M.: Meshfree methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 2, 2 edn., pp. 653–690. John Wiley & Sons, Chichester (2017)
Huerta, A., Fernández-Méndez, S.: Enrichment and coupling of the finite element and meshless methods. Int. J. Numer. Meth. Engng. 48, 1615–1636 (2000)MATHCrossRef
Hughes, T.J.R.: The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, Englewood Cliffs (1987)MATH
Irons, B.M., Zienkiewicz, O.C.: The isoparametric finite element system—a new concept in finite element analysis. In: Proceedings of Conference Recent Advances in Stress Analysis. Royal Aero Soc., London (1968)
Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge (1987)MATH
Kansa, E.J.: Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates. Comput. Math. Appl. 19, 127–145 (1990a)MathSciNetMATHCrossRef
Kansa, E.J.: Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—II solutions to parabolic, hyperbolic and elliptic partial differential equations. Comput. Math. Appl. 19, 147–161 (1990b)MathSciNetMATHCrossRef
Krongauz, Y., Belytschko, T.: Enforcement of essential boundary conditions in meshless approximations using finite elements. Comput. Methods Appl. Mech. Engrg. 131, 133–145 (1996)MathSciNetMATHCrossRef
Larson, M.G., Bengzon, F.: The Finite Element Method: Theory, Implementation, and Applications. Springer, Berlin (2013)MATHCrossRef
Lee, C.K., Liu, X., Fan, S.C.: Local multiquadric approximation for solving boundary value problems. Comput. Mech. 30, 396–409 (2003)MathSciNetMATHCrossRef
Li, S., Liu, W.K.: Meshfree Particle Methods. Springer, Berlin (2004)MATH
Liu, G.R.: Meshfree Methods—Moving Beyond the Finite Element Method, 2nd edn. CRC Press, Boca Raton (2009)CrossRef
Liu, W.K., Chen, Y., Jun, S., Chen, J.S., Belytschko, T., Pan, C., Uras, R.A., Chang, C.T.: Overview and applications of the reproducing Kernel Particle methods. Arch. Comput. Methods Eng. 3, 3–80 (1996)MathSciNetCrossRef
Liu, W.K., Jun, S., Li, S., Adee, J., Belytschko, T.: Reproducing kernel particle methods for structural dynamics. Int. J. Numer. Meth. Engng. 38, 1655–1679 (1995a)MathSciNetMATHCrossRef
Lu, Y.Y., Belytschko, T., Gu, L.: A new implementation of the element free Galerkin method. Comput. Methods Appl. Mech. Engrg. 113, 397–414 (1994)MathSciNetMATHCrossRef
Lucy, L.B.: A numerical approach to the testing of the fission hypothesis. Astron. J. 82, 1013–1024 (1977)CrossRef
Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming, 4th edn. Springer, Cham (2016)MATHCrossRef
McLain, D.H.: Drawing contours from arbitrary data points. Comput. J. 17, 318–324 (1974)CrossRef
Melenk, J.M.: On approximation in meshless methods. In: Blowey, J.F., Craig, A.W. (eds.) Frontiers in Numerical Analysis, pp. 65–141. Springer, Berlin (2005)MATHCrossRef
Melenk, J.M., Babuška, I.: The partition of unity finite element method: Basic theory and applications. Comput. Methods Appl. Mech. Engrg. 139, 289–314 (1996)MathSciNetMATHCrossRef
Moës, N., Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Engng. 46, 131–150 (1999)MathSciNetMATHCrossRef
Moës, N., Dolbow, J. E., Sukumar, N.: Extended finite element methods. In: Stein, E., de Borst, R., Hughes, T.J.R. (eds.) Encyclopedia of Computational Mechanics, vol. 3, 2 edn., pp. 173–193. John Wiley & Sons, Chichester (2017)
Nayroles, B., Touzot, G., Villon, P.: Generalizing the finite element method: Diffuse approximation and diffuse elements. Comput. Mech. 10, 307–318 (1992)MathSciNetMATHCrossRef
Nitsche, J.A.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Semin. Univ. Hambg. 36, 9–15 (1971)MATHCrossRef
Oñate, E., Idelsohn, S., Zienkiewicz, O. C., Taylor, R. L.: A finite point method in computational mechanics. Application to convective transport and fluid flow. Int. J. Numer. Meth. Engng. 39, 3839–3866 (1996)MathSciNetMATHCrossRef
Oñate, E., Perazzo, F., Miquel, J.: A finite point method for elasticity problems. Comput. & Struct. 79, 2151–2163 (2001)CrossRef
Organ, D., Fleming, M., Terry, T., Belytschko, T.: Continuous meshless approximations for nonconvex bodies by diffraction and transparency. Comput. Mech. 18, 225–235 (1996)MATHCrossRef
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2002)MATH
Osher, S., Sethian, J.A.: Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)MathSciNetMATHCrossRef
Rabczuk, T., Zi, G.: A meshfree method based on the local partition of unity for cohesive cracks. Comput. Mech. 39, 743–760 (2007)MATHCrossRef
Randles, P.W., Libersky, L.D.: Smoothed particle hydrodynamics: Some recent improvements and applications. Comput. Methods Appl. Mech. Engrg. 139, 375–408 (1996)MathSciNetMATHCrossRef
Raviart, P.-A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods (Proceedings of Conference, Consiglio Naz. delle Ricerche (C. N. R.), Rome, 1975), Lecture Notes in Mathematics, vol. 606, pp. 292–315. Springer, Berlin (1977)
Ritz, W.: Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. J. Reine Angew. Math. 135, 1–61 (1909)MathSciNetMATHCrossRef
Schröder, J., Schwarz, A., Steeger, K.: Least-squares mixed finite element formulations for isotropic and anisotropic elasticity at small and large strains. In: Schröder, J., Wriggers, P. (eds.) Advanced Finite Element Technologies, pp. 131–175. Springer, Berlin (2016)MATHCrossRef
Schwab, C.: \(p\)- and \(hp\)-Finite Element Methods—Theory and Applications in Solid and Fluid Mechanics. Clarendon Press, Oxford (1998)
Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference, pp. 517–524 (1968)
Simone, A., Duarte, C.A., Van der Giessen, E.: A Generalized Finite Element Method for polycrystals with discontinuous grain boundaries. Int. J. Numer. Meth. Engng. 67, 1122–1145 (2006)MathSciNetMATHCrossRef
Stein, E.: History of the finite element method–mathematics meets mechanics–part I: Engineering developments. In: Stein, E. (ed.) The History of Theoretical, Material and Computational Mechanics-Mathematics meets Mechanics and Engineering, pp. 399–442. Springer, Berlin (2014)MATHCrossRef
Stein, E., Rolfes, R.: Mechanical conditions for stability and optimal convergence of mixed finite elements for linear plane elasticity. Comput. Methods Appl. Mech. Engrg. 84, 77–95 (1990)MathSciNetMATHCrossRef
Stenberg, R.: On some techniques for approximating boundary conditions in the finite element method. J. Comput. Appl. Math. 63, 139–148 (1995)MathSciNetMATHCrossRef
Strouboulis, T., Babuška, I., Copps, K.: The design and analysis of the Generalized Finite Element Method. Comput. Methods Appl. Mech. Engrg. 181, 43–69 (2000)MathSciNetMATHCrossRef
Strouboulis, T., Copps, K., Babuška, I.: The generalized finite element method. Comput. Methods Appl. Mech. Engrg. 190, 4081–4193 (2001)MathSciNetMATHCrossRef
Strutt, J.W. 3rd Baron Rayleigh.: The Theory of Sound, vol. 1. Macmillan and Co., London (1877)
Sukumar, N.: Construction of polygonal interpolants: a maximum entropy approach. Int. J. Numer. Meth. Engng. 61, 2159–2181 (2004)MathSciNetMATHCrossRef
Sukumar, N., Moran, B., Belytschko, T.: The natural element method in solid mechanics. Int. J. Numer. Meth. Engng. 43, 839–887 (1998)MathSciNetMATHCrossRef
Szabó, B., Babuška, I.: Finite Element Analysis. John Wiley & Sons, New York (1991)MATH
Turner, M.J., Clough, R.W., Martin, H.C., Topp, L.J.: Stiffness and deflection analysis of complex structures. J. Aero. Sci. 23, 805–823 (1956)MATHCrossRef
Ventura, G., Xu, J.X., Belytschko, T.: A vector level set method and new discontinuity approximations for crack growth by EFG. Int. J. Numer. Meth. Engng. 54, 923–944 (2002)MATHCrossRef
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)MATH
Wriggers, P.: Nonlinear Finite Element Methods. Springer, Berlin (2008)MATH
Zhu, T., Atluri, S.N.: A modified collocation method and a penalty formulation for enforcing the essential boundary conditions in the element free Galerkin method. Comput. Mech. 21, 211–222 (1998)MathSciNetMATHCrossRef
Zienkiewicz, O.C., Taylor, R.L.: The Finite Element Method, 7th edn. Butterworth-Heinemann, Oxford (2013)MATH
Metadaten
Titel
Galerkin Methods
verfasst von
Marcus Olavi Rüter
Copyright-Jahr
2019
Buch
Error Estimates for Advanced Galerkin Methods

Print ISBN: 978-3-030-06172-2

Electronic ISBN: 978-3-030-06173-9

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-06173-9_4

Premium Partner

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

    Zur Marktübersicht
    Bildnachweise