Research On The Calculation And Engineering Application Of Harmonic Functions And Harmonic Maps In Discrete Situations

Harmonic functions and harmonic mappings are active research fields in the mathematics and engineering.Turning from practice to theory,they connect many fields such as physics,differential equation theory,finite element theory,numerical computation,differential geometry and so on.Due to the simpler algorithms and more efficiency in computation,harmonic functions and harmonic mappings are intensively applied in practice.Based on some classical theoretical results on mathematics,this thesis discusses the computation of harmonic functions and harmonic mappings in the discrete setting,and then applies the computational results into the solutions of some problems in engineering.This thesis studies the following problems:1?Recently,computational conformal mappings are used for a broad range of application in engineering such as geometric modeling,computer graphics,computer vision and so on.Based on the simpler algorithms of harmonic mappings,the first work of this thesis is to approximate the conformal mapping between surfaces with boundaries by harmonic mappings.A harmonic mapping is the critical point of the corresponding integral with respect to the square norm of the gradient or energy density.The harmonic energies defined on Riemann surfaces will decrease along its gradient line directions and reduce to the limit conformal mapping.A harmonic mapping between surfaces is a diffeomorphism which is associated to a unique Beltrami differential.So a sequence of harmonic diffeomorphisms corresponds to a sequence of Beltrami differentials.When the boundary mapping of a surface is restricted on a unit circle in the plane,the sequence of Beltrami differentials changes with constant conformal modulus.This thesis discusses the conformal mapping between surfaces by the sequence of Beltrami differentials with constant conformal modulus,which is equivalent to a sequence of decreasing harmonic energies with fixed boundary correspondences.The thesis provides corresponding algorithms of the numerical computation and proves the convergence of proposed algorithms.At last the thesis demonstrates the numerical experiments for the validity of theoretical argument.2.It is fundamental that the mapping is one to one for all methods applied into parameterizations and shape analysis on computer graphics.In engineering,there is the incomplete research on the property of one to one for discrete harmonic mappings between closed images with complicated topological structures.The second work of this thesis is,in the hyperbolic geometric setting,that applies the idea of a piecewise linear mapping from a planar triangulation to the plane as a hyperbolic convex combination mapping,which considers the image of every interior vertex as a geodesic mass center of the images of its neighbouring vertices.This result can be viewed as a discrete version of the hyperbolic harmonic diffeomorphism between surfaces.Convex combination mappings are more abundant than harmonic mappings.All one to one piecewise linear mappings are convex combination mappings.In the discrete setting,the diffeomorphism means that a hyperbolic harmonic mapping is the one to one piecewise linear mapping on each triangle and preserves the orientation of triangles.In this thesis,a closed discrete surface with high genus can be embedded onto a hyperbolic planar disk by applying Ricci flow method,then the harmonic mapping being one to one can be discussed on the universal covering and one to one on the universal covering means that the hyperbolic harmonic mapping between two closed surfaces with high genus is the diffeomorphism.3.Geometric representations of graphs by contact objects are intensively studied in graph theory and geometry.One of the nicest example is circle contact representations of planar graphs.It asserts that every planar graph can be represented by a set of interiorly disjoint circles such that each vertex is replaced by a circle and an edge is depicted by two circles in contact.Graphs with these geometric representations connect the combinatorial and geometric structure.The case of circles being well studied,it is natural to consider other special representations in mathematics and engineering,for example the square tiling representation has been applied into the combinatorial Riemann mapping theorem.However,there is insufficient results on the computation of square tiling representations of graphs and the existing methods are impractical to compute.The third work of this thesis is to provide a method for computing square tiling representations of planar graphs by combinatorial Hodge theory.The thesis also introduces the theoretical foundation and computational algorithms in details,furthermore this linear method is applied into the square tiling representations of graphs which are embedded on surfaces with high genus.