NCKU Math Colloquium

DATE2016-04-07 16:10-17:00



TITLEApplications of Clifford Analysis to Solid Mechanics and Computational Mechanics

ABSTRACT The theory of complex algebra and complex analysis, such as the holomorphic function theory, Cauchy-Riemann operator, Cauchy-Riemann equation, conformal mapping, Cauchy integral formula, residue theory etc., supports many theoretical and numerical development of solid mechanics in two-dimensional spaces. For Saint-Venant's problem of elasticity, the theory of complex analysis can be used to solve the problems of torsion or flexure. Neglecting the body force, Kolosov and Muskhelishvili successfully used two holomorphic functions to solve a biharmonic equation in a two-dimensional (2D) domain. Considering the problems with body forces, Stevenson used a displacement potential which consists of two holomorphic functions. Without the displacement potential, England adopted one harmonic function and one holomorphic function to represent the displacement field directly. For the plate problems, Goursat's representation of biharmonic functions which adopted the theory of complex analysis can be used to find the solution. Besides isotropic elasticity, the Lekhnitskii formalism and the Stroh formalism had benefited from complex analysis for solving the elastic problems of anisotropic materials. Due to the complexity of the geometry of the domains and of the boundary conditions involved, the development of numerical methods for solving physics problems is necessary. On the basis of the close relationships between holomorphic functions and the Cauchy integral formulae, the boundary element methods (BEMs), whose roots are boundary integral equations (BIEs), are suitable choices. BEMs taking values in complex numbers are called complex variable boundary element methods (CVBEMs). CVBEMs have been applied to solve the general problems of 2D elasticity and also the specific problems such as plane and plate problems, corner and cracks problems, and torsion problems. The remarkable achievement of complex algebra and complex analysis in solving 2D problems has motivated scholars to extend them to deal with problems in three-dimensional (3D) space. However, neither the application of complex algebra and complex analysis to finding solutions in 3D space nor the application to numerical methods has attained the same level of accomplishment as in 2D space. Is there another way to meet the objective? Clifford algebra and Clifford analysis might be the answer. Clifford algebra was introduced by Clifford in 1878. The elements of Clifford algebra, called Clifford numbers or multivectors, allow the operations of addition, subtraction, multiplication, and division. Many authors advocating the application of Clifford algebra renamed it Clifford's geometric algebra and attempted to achieve a standard terminology in physics and mathematics. Moreover, scholars endeavored to develop the theory of multivector calculus and Clifford algebra valued functions, yielding the theory of Clifford analysis. The relation between Clifford algebra and Clifford analysis is similar to that between complex algebra and complex analysis. Hence, Clifford algebra and Clifford analysis might be applied to solid mechanics and computational mechanics to deal with 3D problems. To clarify this point, in this talk, Clifford algebra, its subalgebras, and their analysis would be used to solve 3D problems of elasticity for isotropic and anisotropic materials and also to develop boundary integral equations and boundary element methods.