Clifford algebra valued boundary integral equations and boundary element methods 

Li-Wei Liu

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 mechanics in two-dimensional spaces. 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. The relation between Clifford algebra and Clifford analysis is similar to that between complex algebra and complex analysis. Therefore, Clifford algebra and Clifford analysis might be applied to theoretical mechanics and computational mechanics to deal with 3D problems. To clarify this point, in this short course, Clifford algebra, its subalgebras, and their analysis would be used to solve 3D problems of elasticity and electromagnetics as well as to develop boundary integral equations and boundary element methods. 

 

This course is designed for the beginner of Clifford algebra valued boundary integral equations and boundary element methods such as young researchers and graduate students.