Kevin Cole

Contact Information:

City Campus (Lincoln)

Professor Emeritus

Academic Degrees

  • Ph.D., 1986 Michigan State University
  • M.S., 1979 University of Minnesota
  • B.S., 1977 Iowa State University

Areas of Research and Professional Interest

  • Additive Manufacturing
  • Analytical Methods
  • Heat Transfer and Mass Diffusion
  • Interpretation of Experimental Data
  • Inverse Problems
  • Thermal/fluid sensors


  1. R. Yavari, P. K. Rao, K. D. Cole, Simulating heat flux in additive manufacturing, No. 17/275,735, 24 Feb 2022.
  2. J. Severson, P. K. Rao, R. Yavari, K. D. Cole, Thermal modeling of additive manufacturing using graph theory,  NO. 17/499,402, 14 April 2022.
  3. R. Yavari, P. K. Rao, K. D. Cole, Thermal modeling of additive manufacturing using progressive horizontal subsections, No. 17/668,025, 8 Sept 2022.
  4. R. Yavari, P. Rao, A. Riensche, K. D. Cole, Systems and methods for combining thermal simulations with sensor data to detect flaws and malicous cyber intrusions in additive manufacturing, WO 2023/03884, 16 March 2023.
  5. K. D. Cole, A. Riensche, P. Rao, Thermal modeling of additive manufacturing, No. WO 2023/076636, 4 May 2023.

About Kevin D. Cole

Dr. Cole studies heat and mass transfer through analytical and semi-analytical methods, with application to manufacturing processes, inverse problems, and measurement of thermal properties. Dr. Cole retired from teaching and tie-wearing in 2020 to focus on research and software commercialization. He has current funding from the Department of Energy and the National Science Foundation. He is co-author of a reference book on heat conduction and maintains a website called the Green's Function Library.

He led a team that developed the 
Exact Analytical Conduction Toolbox funded by the National Science Foundation. Dr. Cole is a member of the American Society of Mechanical Engineers (ASME), American Institute of Aeronautics and Astronautics (AIAA), and is a registered professional engineer.

Selected Publications


  • Cole, K. D., Beck, J. V., Haji-Sheikh, A., and Litkouhi, B., Heat Conduction Using Green's Functions, 2nd edition, Taylor and Francis, New York, 2011.


  • A. Riensche, B. D. Bevans, Z. Smoqi, R. Yavari, A. Krishnan, J. Gilligan, N. Piercy, K. D. Cole, P. K. Rao, Feedforward Control of Thermal History in Laser Powder Bed Fusion: Toward Physics-based Optimization of Processing Parameters, Materials and Design, v. 224, December 2022, 111351,
  • A. Riensche, J. Severson, R. Yavari, N. L. Piercy, K. D. Cole, P. K. Rao, “Thermal Modeling of Directed Energy Deposition Additive Manufacturing using Graph Theory,” Rapid Prototyping Journal, vol. 29, no. 2, pp. 324-343, 2022,
  • Cole, K. D., Riensche, A., Rao, P. K. "Discrete Green's functions and spectral graph theory for computationally efficient thermal modeling ", Int. J. Heat Mass Transfer, vol. 183, part B, February 2022, 122112.
  • Yavari, R., Riensche, A., Tekerek, E.,  Jacquemetton, L., Halliday, H. Vandever, M., ; Tenequer, A., Perumal, V., Kontsos, A., Smoqi, Z, Cole, K. D., Rao, P. K., "Digitally Twinned Additive Manufacturing: Real-time Detection of Flaws in Laser Powder Bed Fusion by Combining Thermal Simulations with In-Situ Meltpool Sensor Data," Materials & Design, Vol 211, 110167,, 2021.
  • Cole, K. D., "To engineer is human: gender issues in the engineering workplace," Torch Magazine, vol. 92, no. 2, pp. 10-14, 2021.
  • Ҫetin, Barbaros., Kuşcua, Y. F., Ҫetin, Bariş, Tümüklüd, O., Cole, K. D., "Semi-analytical source (SAS) method for 3-D transient heat conduction problems with moving heat source of arbitrary shape," Int. J. Heat Mass Transfer, v. 165,, 2021.
  • Yavari, R., Smoqi, Z., Bevans, B., Kobir, H., Mendoza, H. Song, H., Cole, K. D., Rao, P., Part-scale thermal simulation of laser powder bed fusion using graph theory: effect of thermal history on porosity, microstructure, and recoater-crash, Materials & Design, vol. 204, 109685,, 2021.
  • Yavari, R., Williams, R. J., Riensche, A. Hooper, P. A., Cole, K. D., Jacquemetton, L., Halliday, H., Rao, P., Thermal modeling in metal additive manufacturing using graph theory – application to laser powder bed fusion of a large volume impeller, Additive Manufacturing, Vol. 41, 101956,, 2021.
  • Yavari, R., Williams, R. J., Cole, K. D., Hooper, P. A., and Rao, P., "Thermal Modeling in Metal Additive Manufacturing using Graph Theory: Experimental Validation with In-situ Infrared Thermography Data from Laser Powder Bed Fusion." ASME. J. Manuf. Sci. Eng. doi: 10.1115/1.4047619, 2020.
  • Gaikwad, A., Yavari, M. R., Montazeri, M., Cole, K. D., Bian, L. K., Rao, P. K., "Toward the digital twin of additive manufacturing: Integrating thermal simulations, sensing, and analytics to detect process faults," IISE Transactions,, 2020.
  • Pi, T., Cole, K. D., Zhao, Q., Zhao, W., "Investigation of Numerical Evaluation Improvement for Three-Dimensional Infinite Cylindrical Heat Conduction Problems," J. Heat Transfer, 142(4): 044501,, 2020.
  • Cole, K. D., Yavari, M. R., Rao, P., "Computational heat transfer with spectral graph theory: Quantitative verification," Int. J. Thermal Sciences, Vol. 153, 10638, 2020.

  • Yavari, M. R., Cole, K. D., Rao, P., "Thermal modeling in metal additive manufacturing using graph theory," J. Manufacturing Science and Engineering, 141 (7), 071007 (May 21, 2019) doi: 10.1115/1.4043648, 2019.

  • Cole, K. D., Cetin, B., Demirel, Y., “Semi-analytical source method for reaction-diffusion problems,” J. Heat Transfer, doi:10.1115/1.4038987, 2018.
  • Pi, T., Cole, K. D., Beck, J. V., “Efficient numerical evaluation of exact solutions for 1D and 2D infinite cylindrical heat conduction problems,” J. Heat Transfer, vol. 139, pp 121301.1-10,  2017.
  • Cole, K. D., Cetin, B., “Modeling of Joule Heating and Convective Cooling in a Thick-walled Micro-tube,” Int. J. Thermal Sciences, vol. 119, pp. 24–36, 2017.
  • Cole, K. D, Beck, J. V., Woodbury, K. A. de Monte, F., "Intrinsic verification and a heat conduction database," Int. J. Thermal Sciences, vol. 78, pp. 36 – 47, 2014.
  • Cole, K. D., Cetin, B., and Brettmann, L., "Microchannel heat transfer with slip flow and wall effects," AIAA J. Thermophysics and Heat Transfer, vol 28, No. 3, pp. 455-462, 2014.
  • Tian, T., and Cole, K. D., “Anisotropic Thermal Conductivity Measurements of Carbon-fiber/epoxy Composites," Int. J. Heat Mass Transfer, vol. 55 pp. 6530 – 6537, 2012.
  • Cole, K. D., and Cetin, B., "The Effect of Axial Conduction on Heat Transfer in a Liquid Microchannel Flow," Int. J. Heat Mass Transfer, vol. 54, pp. 2542-2549, 2011.


  • Cole, K. D., R02B5Z00T0 Infinite solid cylinder heated over half of its surface and zero initial temperature, Exact Analytical Conduction Toolbox, Dec. 20, 2023,
  • Cole, K. D., R00Z20B5T0 Semi-infinite body with a disk-shaped surface heat source, Exact Analytical Conduction Toolbox, Dec. 20, 2023,
  • Cole, K. D., R03B0Z33B00T5 Finite cylinder with convective boundaries and

    piecewise initial condition, Exact Analytical Conduction Toolbox, Nov. 15, 2023,

  • Cole, K. D., phi22B10F0T0 Thin cylindrical shell initially at ambient temperature heated at one end and cooled along its circumference, Exact Analytical Conduction Toolbox, Nov. 2, 2023,
  • Cole, K. D., R21B00T- Hollow cylinder insulated at the minor radius and zero temperature at the major radius

    with logarithmic initial temperature, Exact Analytical Conduction Toolbox, Oct. 17, 2023,

  • Cole, K. D., R11B00T1 Hollow cylinder with zero temperature boundaries and uniform initial temperature, Exact Analytical Conduction Toolbox, Oct. 10, 2023,