The method of finite spheres in acoustic wave propagation through nonhomogeneous media: Inf-sup stability conditions

Williams L. Nicomedes, Klaus-Jürgen Bathe, Fernando J. S. Moreira, Renato C. Mesquita

Abstract


When the method of finite spheres is used for the solution of time-harmonic acoustic wave propagation problems in nonhomogeneous media, a mixed (or saddle-point) formulation is obtained in which the unknowns are the pressure fields and the Lagrange multiplier fields defined at the interfaces between the regions with distinct material properties. Then certain inf-sup conditions must be satisfied by the discretized spaces in order for the finite-dimensional problems to be well-posed. We discuss in this paper the analysis and use of these conditions. Since the conditions  involve norms of functionals in fractional Sobolev spaces, we derive ‘stronger’ conditions that are simpler in form. These new conditions pave the way for the inf-sup testing, a tool for assessing the stability of the discretized problems.


Keywords


acoustic waves; finite elements; finite spheres; inf-sup conditions; meshfree methods

Full Text:

PDF

References


S. De and K. J. Bathe. The method of finite spheres. Computational Mechanics, 25, (4), (2000), pp. 329–345. https://doi.org/10.1007/s004660050481.

J. J. Monaghan. An introduction to SPH. Computer Physics Communications, 48, (1988), pp. 89–96.

T. Belytschko, Y. Y. Lu, and L. Gu. Element-free Galerkin methods. International journal for numerical methods in engineering, 37, (2), (1994), pp. 229–256. https://doi.org/10.1002/nme.1620370205.

S. Atluri and S. Shen. The meshless local Petrov-Galerkin method: A simple and less-costly alternative to the finite-element and boundary element methods. CMES: Computer Modeling in Engineering & Sciences, 3, (2002), pp. 11–51.

G. R. Liu. Meshfree methods: moving beyond the finite element method. 2nd edition, CRC Press, (2010).

K. J. Bathe. Finite element procedures. Prentice Hall, (1996). 2nd edition, K. J. Bathe, Watertown MA, (2014). Higher Education Press China, (2016).

K. J. Bathe and L. Zhang. The finite element method with overlapping elements–a new paradigm for CAD driven simulations. Computers & Structures, 182, (2017), pp. 526–539. https://doi.org/10.1016/j.compstruc.2016.10.020.

L. Zhang, K. T. Kim, and K. J. Bathe. The new paradigm of finite element solutions with overlapping elements in CAD–Computational efficiency of the procedure. Computers & Structures, 199, (2018), pp. 1–17. https://doi.org/10.1016/j.compstruc.2018.01.003.

W. L. Nicomedes, K. J. Bathe, F. J. S. Moreira, and R. C. Mesquita. Meshfree analysis of electromagnetic wave scattering from conducting targets: Formulation and computations. Computers & Structures, 184, (2017), pp. 36–52. https://doi.org/10.1016/j.compstruc.2017.01.014.

W. L. Nicomedes, K. J. Bathe, F. J. S. Moreira, and R. C. Mesquita. Mesquita, Acoustic scattering in nonhomogeneous media and the problem of discontinuous gradients: Analysis and inf-sup stability in the method of finite spheres. submitted, (2020).

F. B. Jensen,W. A. Kuperman, M. B. Porter, and H. Schmidt. Computational ocean acoustics. 2nd edition, Springer, (2011).

F. Brezzi and M. Fortin. Mixed and hybrid finite element methods. Springer, (1991).

D. Boffi, F. Brezzi, and M. Fortin. Mixed finite element methods and applications. Springer, (2013).

Y. Chai and K. J. Bathe. Transient wave propagation in inhomogeneous media with enriched overlapping triangular elements. Computers & Structures, 237, (2020). https://doi.org/10.1016/j.compstruc.2020.106273.

L. Tartar. An introduction to Sobolev spaces and interpolation spaces. Springer, (2007).

G. Leoni. A first course in Sobolev spaces. 2nd edition, American Mathematical Society, (2017).

E. DiBenedetto. Real analysis. 2nd edition, Birkh¨auser, (2016).

C. Farhat, I. Harari, and U. Hetmaniuk. A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Computer Methods in Applied Mechanics and Engineering, 192, (11-12), (2003), pp. 1389–1419. https://doi.org/10.1016/s0045-7825(02)00646-1.

I. Kalashnikova, C. Farhat, and R. Tezaur. A discontinuous enrichment method for the finite element solution of high Péclet advection–diffusion problems. Finite elements in Analysis and Design, 45, (4), (2009), pp. 238–250. https://doi.org/10.1016/j.finel.2008.10.009.

E. Grosu and I. Harari. Three-dimensional element configurations for the discontinuous enrichment method for acoustics. International Journal for Numerical Methods in Engineering, 78, (11), (2009), pp. 1261–1291. https://doi.org/10.1002/nme.2525.

C. Farhat, I. Kalashnikova, and R. Tezaur. A higher-order discontinuous enrichment method for the solution of high péclet advection–diffusion problems on unstructured meshes. International Journal for Numerical Methods in Engineering, 81, (5), (2010), pp. 604–636. https://doi.org/10.1002/nme.2706.

I. Kalashnikova, R. Tezaur, and C. Farhat. A discontinuous enrichment method for variable-coefficient advection-diffusion at high P´eclet number. International Journal for Numerical Methods in Engineering, 87, (1-5), (2010), pp. 309–335. https://doi.org/10.1002/nme.3058.

F. Brezzi and K. J. Bathe. A discourse on the stability conditions for mixed finite element formulations. Computer Methods in Applied Mechanics and Engineering, 82, (1-3), (1990), pp. 27–57. https://doi.org/10.1016/0045-7825(90)90157-h.

N. El-Abbasi and K. J. Bathe. Stability and patch test performance of contact discretizations and a new solution algorithm. Computers & Structures, 79, (16), (2001), pp. 1473–1486. https://doi.org/10.1016/s0045-7949(01)00048-7.

N. Moes, E. Bechet, and M. Tourbier. Imposing Dirichlet boundary conditions in the extended finite element method. International Journal for Numerical Methods in Engineering, 67, (12), (2006), pp. 1641–1669. https://doi.org/10.1002/nme.1675.

E. Bechet, N. Moes, and B. Wohlmuth. A stable Lagrange multiplier space for stiff interface conditions within the extended finite element method. International Journal for Numerical Methods in Engineering, 78, (8), (2009), pp. 931–954. https://doi.org/10.1002/nme.2515.

G. Ferte, P. Massin, and N. Moes. Interface problems with quadratic X-FEM: design of a stable multiplier space and error analysis. International Journal for Numerical Methods in Engineering, 100, (11), (2014), pp. 834–870. https://doi.org/10.1002/nme.4787.

K. J. Bathe. The inf–sup condition and its evaluation for mixed finite element methods. Computers & Structures, 79, (2), (2001), pp. 243–252. https://doi.org/10.1016/s0045-7949(00)00123-1.

D. Chapelle and K. J. Bathe. The inf-sup test. Computers & Structures, 47, (4-5), (1993), pp. 537–545. https://doi.org/10.1016/0045-7949(93)90340-j.

W. Bao, X. Wang, and K. J. Bathe. On the inf–sup condition of mixed finite element formulations for acoustic fluids. Mathematical Models and Methods in Applied Sciences, 11, (05), (2001), pp. 883–901. https://doi.org/10.1142/s0218202501001161.

A. F. Peterson, S. L. Ray, R. Mittra, I. of Electrical, and E. Engineers. Computational methods for electromagnetics. IEEE Press Series on Electromagnetic Waves, (1998).

L. C. Evans. Partial differential equations. 2nd edition, American Mathematical Society, (2010).

J. T. Oden and J. N. Reddy. An introduction to the mathematical theory of finite elements. Dover, (2011).

S. Salsa. Partial differential equations in action: from modelling to theory. 3rd edition, Springer, (2016).

W. McLean. Strongly elliptic systems and boundary integral equations. Cambridge University Press, (2000).

J. C.Nedelec. Acoustic and electromagnetic equations: integral representations for harmonic problems. Springer, (2001).

D. Colton and R. Kress. Inverse acoustic and electromagnetic scattering theory. 3rd edition, Springer, (2013).

V. Girault and P. A. Raviart. Finite element methods for Navier-Stokes equations: theory and algorithms. Springer, (1986).

A. Ern and J. L. Guermond. Theory and practice of finite elements. Springer, (2004).

F. Boyer and P. Fabrie. Mathematical tools for the study of the incompressible Navier-Stokes equations and related models. Springer, (2012).

D. Chapelle and K. J. Bathe. The finite element analysis of shells-fundamentals. 2nd edition, Springer, (2011).

P. G. Ciarlet. Linear and nonlinear functional analysis with applications. Society for Industrial and Applied Mathematics – SIAM, (2013).

D. Chapelle and K. J. Bathe. On the ellipticity condition for model-parameter dependent mixed formulations. Computers & Structures, 88, (9-10), (2010), pp. 581–587.

F. Ihlenburg. Finite element analysis of acoustic scattering. Springer, (1988).

A. Moiola and E. A. Spence. Is the Helmholtz equation really sign-indefinite? SIAM Review, 56, (2), (2014), pp. 274–312. https://doi.org/10.1137/120901301.

K. J. Bathe, A. Iosilevich, and D. Chapelle. An inf-sup test for shell finite elements. Computers & Structures, 75, (5), (2000), pp. 439–456. https://doi.org/10.1016/s0045-7949(99)00213-8.

Y. Ko and K. J. Bathe. Inf-sup testing of some three-dimensional low-order finite elements for the analysis of solids. Computers & Structures, 209, (2018), pp. 1–13. https://doi.org/10.1016/j.compstruc.2018.07.006.

P. Grisvard. Elliptic problems in nonsmooth domains. Society for Industrial and Applied Mathematics – SIAM, (2011). https://doi.org/10.1137/1.9781611972030.

J. R. Munkres. Topology. 2nd edition, Pearson, (2000).




DOI: https://doi.org/10.15625/0866-7136/15336 Display counter: Abstract : 99 views. PDF : 32 views.

Refbacks

  • There are currently no refbacks.


Copyright (c) 2020 Vietnam Academy of Science and Technology


                       


Editorial Office of Vietnam Journal of Mechanics

3rd Floor, A16 Building, 18B Hoang Quoc Viet Street, Cau Giay District, Hanoi, Vietnam
Tel: (+84) 24 3791 7103
Email: vjmech@vjs.ac.vn