Topology Optimization Using Load Path and Homogenization


Kaveh Gharibi

Date of Award


Document Type

Dissertation - Open Access

Degree Name

Doctor of Philosophy in Aerospace Engineering


Aerospace Engineering

Committee Chair

Dr. Ali Tamijani

First Committee Member

Dr. Susan Davis Allen

Second Committee Member

Dr. Habib Eslami

Third Committee Member

Dr. Harwan Al-Haik


In this work, the connection between topology optimization and load transfer has been established. New methods for determining load paths in two dimensional structures, plates and shells are introduced. In the two-dimensional space, there are two load paths with their total derivative equal to the transferred load, their partial derivatives related to stress tensor, and satisfying equilibrium. In the presence of a body load the stress tensor can be decomposed into solenoidal and irrotational fields using Gurtin or Helmholtz decomposition. The load path is calculated using the solenoidal field. A novel method for topology optimization using load paths and total variation of different objective functions is formulated and implemented. This approach uses the total variation to minimize different objective functions, such as compliance and norm of stress subjected to equilibrium. Since the problems are convex, the optimized solution is a global optimum which is found by solving the Euler-Lagrange optimality criteria. The optimal density of a structure is derived using optimality criteria and optimized load paths. To attain the topology of the microstructure, the principal load paths that follow the optimal principal stress directions are calculated. Since the principal stress vector field is not curl free, a dilation field is multiplied to extract the curl free component of principal stress vectors. The principal vector field has singularities which are removed by an interpolation scheme that rotates the vectors by n to construct a coherent vector field. The optimal periodic rectangular microstructure is constructed using the load functions and microstructure dimensions. The advantage of this scheme is that using the load path reduces the equilibrium constraints from two to one, and the variables are reduced from three stresses to two load functions. The non-linear elliptic partial differential equations which are derived from the total variation equations (Euler-Lagrange) are solved using the Gauss- Newton method which has a quadratic convergence, speeding up the convergence towards the optimal structure.


The author wishes for only the abstract of the work to be available at this time.