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      "statement": "# Courant: approximate deformation with a flexible triangular mesh\n\nCourant described variational approximations using continuous piecewise-linear functions on flexible triangular nets and demonstrated them on planar torsion problems, an early step toward finite-element methods.\n\nTo estimate how a membrane or structure deforms, one can minimize an energy rather than solve the differential equation directly. Rayleigh and Ritz had already developed this approach. Courant’s 1943 address described using functions that are linear inside each triangle of a mesh, with their values at the mesh points as unknowns. The energy integral then becomes a finite calculation and minimization gives linear equations. Flexible triangular meshes can fit the shape of the domain better than a rectangular grid or a few global formulas. An appendix applied this idea to the torsional stiffness of square sections with holes. The paper also warned that a good energy value does not automatically give an accurate local deformation or its derivatives, especially with very few elements. Higher-derivative problems require additional admissibility and were left for separate extension. This entry credits the scoped triangular-net construction and examples, preserving earlier variational methods and later finite-element engineering as distinct contributions.\n\n## Scientific scope\n\nThe triangular trial functions are continuous and piecewise linear, suitable for the stated variational energies involving first derivatives and appropriate boundaries. They are not a universal conforming basis for higher-derivative plate problems or every PDE. Existence/minimum-attainment assumptions remain necessary. The contribution is distinct from all later finite-element convergence and mesh-quality theory.\n\n## People and systems\n\nR. Courant\n\n\n\n## Original sources\n\n- [Courant, Variational Methods for Equilibrium and Vibrations,1943author text in1994reprint2165-2187,Stanford course copy](https://web.stanford.edu/class/cme324/classics/courant.pdf)\n- [AMS publisherCrossrefdepositfor1943original](https://api.crossref.org/works/10.1090%2FS0002-9904-1943-07818-4)\n- [Wiley metadata forFelippa1994appreciation andclassicreprint composite](https://api.crossref.org/works/10.1002%2Fnme.1620371303)\n",
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