Andrea Dziubek

Research Interests

• Structure Preserving Numerical Methods, Discrete Exterior Calculus and Finite Element Methods
• Theoretical Mechanics and Differential Geometry (Elasticity, Shells, Fluids, Materials)
• Modeling and Simulation of Problems in Ophthalmology (PDEs on curved Surfaces)

Teaching

• MAT 460 Numerical Differential Equations - Student Projects: Posters and Reports
• Applied Mathematics and Physics Courses
• Advanced Certificate Computational Mathematical Modeling (CMM)
• SUNY Poly Stories (Mathematics)
• EyeDEC SURP
• Python Bootcamp Jupyter Notebooks: P1.ipynb, P2.ipynb, P3.ipynb.
• For Course Information see D2L

Selected Publications

• Intrinsic mixed finite element methods for linear Cosserat elasticity and couple stress problem. Andrea Dziubek, Kaibo Hu, Michael Karow, Michael Neunteufel. Submitted. 10/2024
  arXiv:2410.14176v1
  We propose two parameter-robust mixed finite element methods for linear Cosserat elasticity. The Cosserat coupling constant $\mu_c$, connecting the displacement $u$ and rotation vector $\omega$, leads to possible locking phenomena in finite element methods. The formal limit of $\mu_c\rightarrow\infty$ enforces the constraint $\tfrac12\text{curl}\,u=\omega$ and leads to the fourth order couple stress problem. Viewing the linear Cosserat model as the Hodge-Laplacian problem of a twisted de Rham complex, we derive structure-preserving distributional finite element spaces, where the limit constraint is fulfilled in the discrete setting. Applying the mass conserving mixed stress (MCS) method for the rotations, the resulting scheme is robust in $\mu_c$. Combining it with the tangential-displacement normal-normal-stress (TDNNS) method for the displacement part we obtain additional robustness in the nearly incompressible regime and for anisotropic structures. Using a post-processing scheme for the rotations, we prove optimal convergence rates independent of the Cosserat coupling constant $\mu_c$. Further, we propose a mixed method for the couple stress problem based on the MCS scheme. We demonstrate the performance of the proposed methods in several numerical benchmark examples.
• Tensor Analysis, Heinz Schade, Klaus Neemann, translated by Andrea Dziubek, Edmond Rusjan. De Gruyter. 2018
  ISBN 978-3-11-040426-5 (German version, goodreads).
  Contents: Algebraic Tools, Tensor Analysis in Symbolic Notation and in Cartesian Coordinates, Algebra of Second Order Tensors, Tensor Analysis in Curvilinear Coordinates, Representation of Tensor Functions. Appendices: Solutions to the Problems; Cylindrical Coordinates and Spherical Coordinates. The book contains numerous exercises with answers, making it suitable for self study for students and researchers in areas such as solid mechanics, fluid mechanics, and electrodynamics. The German version is in its 3rd revised edition.
• Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model. A. Dziubek, G. Guidoboni, A. Harris, A.H. Hirani, E. Rusjan, W. Thistleton. Biomechanics and Modeling in Mechanobiology, pp 1-15, Springer, DOI: 10.1007/s10237-015- 0731-8. 2015
  (abstract,pdf)
• Using Generalized Dimensional Analysis to Obtain Reduced Effective Model Equations for Condensation in Slender Tubes with Rotational Symmetry. ASME Journal of Heat Transfer, 135(5), 051501. 2013
  arxiv:1101.5718
• Equations for Two-Phase Flows: A Primer. Meccanica, Vol 47(8), pp 1819-1836. 2012
  arxiv:1101.5732
• Modeling, Analysis and Numerical Simulation of a Moving Boundary Problem with Phase Change and Surface Tension. Thesis, Berlin University of Technology, Germany, 12/2006
• Lecture Notes: Bilanzgleichungen (Introduction to Continuum Mechanics Script, German)

Research Projects

• Geometric Mechanics and Structure Preserving Numerical Methods of (Cosserat) Shell Elasticity: Many phenomena in physical, engineering, and biological systems are modeled by elastic shells. In contrast to classical continuum theory, where the smallest constitutive element of media is the point-particle, in Cosserat theory translations and rotations have a similar status. Cosserat shell models could be the key in understanding chain-complexes - and chain-complexes are fundamental for deriving structure preserving numerical methods. [International collaborators and workshops]
• Exterior Calculus and Finite Element Exterior Calculus: We model problems from science and engineering in exterior calculus language and solve them using the mixed methods of FEEC. We implement and analyze solution algorithms in NGSolve, a modern FE software package for partial differential equations on complex geometries. The goal is to demonstrate the potential of these methods for teaching and in applications. [Undergraduate research]
• Combining Image Analysis and Blood Flow Simulation: Geometric reconstruction of vascular tree networks is a critical step in the development of a physically based patient specific blood flow analysis framework. To accurately extract vascular networks from patient data, advanced image segmentation and classification approaches are necessary. Together with M. Reale and E. Rusjan (SUNY Poly) we combine image analysis and blood flow simulation in an end-to-end system to determine the resulting tree's feasibility. [SUNY POLY collaboration]
• Ocular Blood Flow Modeling (previous): Together with G. Guidoboni, (Electrical and Computer Engineering, University of Maine), A. Hirani, (Math, University of Illinois) and E. Rusjan (SUNY Poly), we developed a computational model to theoretically investigate retinal perfusion, i.e., hemodynamics in the microvasculature nourishing the retinal tissue. The model predictions support the clinical hypothesis that alterations in ocular shape, such as those occurring in myopic eyes, might be associated with pathological alterations in retinal hemodynamics. [Previous work]

Research Activites, SUNY Poly

• Utica Fall Workshop 2023 Geometric Mechanics and Structure Preserving Discretizations of Shell Elasticity
  
  July 17 - July 28, 2023: A. Dziubek, M. Karow, M. Neunteufel, K. Hu

  
  2 talks by Michael Neunteufel, Analysis and Scientific Computing, TU Wien, Austria, co-developer of NGSolve
  

  • Combining (discrete) differential geometry with finite elements to develop structure preserving numerical methods for continuum mechanics (shell elasticity)
  • Using Netgen/NGSolve to quickly translate models from solid mechanics, fluid dynamics and electromagnetic into efficient finite element code

  (Suny Poly News)

• Utica Fall Workshop 2019 Geometric Mechanics and Structure Preserving Discretizations of Shell Elasticity
  August 27 - September 2, 2019: A. Dziubek, M. Karow, R. Lowry, S. Sadik, C. Stoica, E. Rusjan

These two workshops are follow-up workshops to the 2015 Workshop

• Seminar Series on Shell Models
  • Shells I: Differential Geometry, Andrea Dziubek
    30 September, 2016, noon, DON 2159
  • Shells II: Elements of Functional and Numerical Analysis, Kaushik Kalyanaraman, SCOREC, RPI, NY
    4 November 2016, noon, DON 2159
  • Shells III: Mathematical Shell Models, Andrea Dziubek
    9 December 2016, noon, KUN A131 (Utica), NFN 203 (CNSE, Albany)
  • Shells IV: On the Moving Frame Method in Differential Geometry of Surfaces in R3 and its Application to Shell Theory, Wenfeng Chen
    4 February 2017, 1pm, DON 2159
  • Shells V: Notes on Connections and Forms, Michael Karow, TU-Berlin (Germany)
    21 September 2018, 1pm, DON 2159
• Workshop on Geometry and Symmetry based Mathematical and Computational Methods with Applications in Engineering, Science and Education (Workshop Webpage)
  
  August 24 - August 30, 2015: A. Dziubek, A. Hirani, A. Angoshtari, C. Cafaro, C. Stoica, E. Rusjan, F. Valiquette, H. Heumann, K. Kalyanaraman, M. Mohamed, M. Karow, R. Lowry, T. Schmah, T. Sadigov, W. Chen, W. Bauer, Z. Thomova, D. Yaciuk, M. Yermakov

Undergraduate Research Activities, SUNY Poly

• FEEC - Independent Study (Fall 2023) and SUNY Undergraduate Research Program (Summer 2024)
  Student Poster Presentations (in addition to SUNY Poly Showcase Day Posters)

  • WCCM24 - 16th World Congress on Computational Mechanics, Vancouver, Canada, 6/21-26 2024
    Geometry and Topology in New Numerical Methods for Electromagnetism and Continua, Nicholas Andrzejkiewicz
  • APS - Topical Symposium of the New York State Section of the American Physical Society Fall 2024: Advances in Computational Physics and the Physics of Computation, SUNY Polytechnic, Utica, 10/26, 2024
    A Glance at Finite Element Analysis and Exterior Calculus Chawn Neal (First Poster Price)

  (SUNY Poly Math REU Summer '24 Story)
• Independent Study on Finite Element Exterior Calculus, Undergraduate Research, Fall 2023
• SUNY Summer Undergraduate Research Program, 2021, 2019, 2018, 2014, 2015
• Slocum Dickson Research Award Scholarship, August 2015 - May 2016
• Hudson River Undergraduate Mathematics Conference, 2015, 2013, 2012
• Computational Boot Camp, June 15 - June 26, 2015
  (SUNY Poly Bootcamp Story)

Recent Presentations

• SNP - Nonlinear Theories in Mechanics, 58th Meeting of the Society for Natural Philosophy, Department of Mechanical and Production Engineering, Aarhus University, Denmark, 6/12/2024
  Intrinsic Mixed Finite Element Methods for Linear Cosserat Elasticity, A. Dziubek (round-table, speaker), joint work with M. Karow, M. Neunteufel, K. Hu
• INI - Geometry, compatibility and structure preservation in computational differential equations (GCSW01), Isaac Newton Institute, Cambridge, UK, 7/8-12, 2019
  Poster: Shell Models I -Cheat Sheet- Cosserat, Reissner-Mindlin, Koiter, Kirchhoff-Love, A. Dziubek (presenter), A. Gregory, M. Karow, E. Rusjan
• CMS - Winter meeting, Advances and Applications in Geometric and Structure-Preserving Discretizations Session, Vancouver, Canada, 12/07-10, 2018
  Literature Review of Structure Preserving Discretizations for Shell Models
• IMA - Institute for Mathematics and its Applications, Special Workshop Mathematics and Mechanics in the 22nd Century: Seven Decades and Counting..., Eugene, OR, 10/23-25, 2015
  Poster: Effect of ocular shape and vascular geometry on retinal hemodynamics, A. Dziubek (presenter), joint work with G. Guidoboni, A. Harris, A. N. Hirani, E. Rusjan, W. Thistleton
• SEI - 7th Annual SUNY Eye Institute Meeting, SUNY POLY, CNSE, Albany, NY, 9/26/2015
  Poster: Recent Progress in the EYE DEC Retinal Perfusion Model - Changes in Ocular Shape play a role in Retinal Blood Perfusion, A. Dziubek, A. Harris, A.N. Hirani, E. Rusjan, W. Thistleton.
• IMA - Institute for Mathematics and its Applications, Special Workshop Structure-Preserving Discretizations of Partial Differential Equations, University of Minnesota, MN, 10/22-24, 2014
  Poster: DEC - A Software Environment to Model Blood Flow in the Eye, A. Dziubek, E. Rusjan
• SEI - 6th Annual SUNY Eye Institute Meeting, Upstate Medical University Syracuse, NY, 9/27/2014
  How to Validate a Mathematical Model of the Eye, Joint work with A. Benavente-Perez (SUNY College of Optometry, NYC), E. Rusjan, B. Thistleton
• MAA Math Fest, Portland, Oregon, Invited Paper Session, The Eyes Have It: Mathematical Modeling of the Retina, 8/9/2014
  New Paradigms in Retinal Blood Flow Simulation (Invited by E. Camacho)
• WCCM XI, ECCM V, ECFD VI, 11th. World Congress on Computational Mechanics, Barcelona, Spain, Minisymposium Computational Modeling of Multiphysics/Multiscale Coupled Processes in Biological and Nanotechnological Systems, 7/25/2015
  Retinal blood flow changes and vascular parameters and structure, A. Dziubek (speaker), E. Rusjan and W. Thistleton. (Invited by R. Sacco, POLIMI, Italy, G. Guidoboni, IUPUI, R. Melnik, WLU, Canada)
• COPDE - Conference on Partial Differential Equations, Novacella, Italy, 5/30/2014
  Retinal blood flow changes and vascular parameters and structure, A. Dziubek, E. Rusjan (speaker), W. Thistleton. (Invited talk)
• SEI - 5th Annual SUNY Eye Institute Meeting, Downstate Medical Center, Brooklyn, NY, 10/19-20, 2013
  Presented the ARVO and FMD posters
• FMD - Proceedings of the ASME/FDA 2013, 1st Annual Frontiers in Medical Devices, Washington DC, 12 September 2013
  Poster: Challenges in Blood Flow Simulation: Numerical Methods and Image Processing Tools, A. Dziubek, E. Rusjan, W. Thistleton
  
• AMMCS - Applied Mathematics, Modeling and Computational Science, Waterloo, August 28, 2013
  Modeling Oxygen Dynamics of the Retina - Using Discrete Exterior Calculus, A. Dziubek, G. Guidoboni, A. Hirani, E. Rusjan (speaker)
  
• 8th Conference on Applied Mathematics and Scientific Computing, Sibenik, Croatia, June 12, 2013
  A Discrete Exterior Calculus Model of Blood Flow in the Retina, A. Dziubek, G. Guidoboni, A. Hirani, E. Rusjan (speaker)
  
• ARVO - The Association for Research in Vision and Ophthalmology, Seattle, 6 May 2013
  Poster: Effect of ocular curvature and myopia on retinal blood flow: a theoretical study, A Dziubek, G. Guidoboni, A. Hirani, A. Harris
  
• AMS Sectional Meeting, Rochester Institute of Technology, NY, September 22-23, 2012, Special Session on Applied and Computational Mathematics (Ludwig Kohaupt)
  Modeling the oxygen dynamics of the retina of the eye with discrete exterior calculus, A. Dziubek, G. Guidoboni, A. Hirani, E. Rusjan
  Optimizing Discrete Exterior Calculus, E. Rusjan, A. Dziubek
• Focus Program on Geometry, Mechanics and Dynamics, the Legacy of Jerry Marsden, Fields Institute, Toronto, CA, July 2012 (D.E. Chang, D. Holm, G.W. Patrick, T. Ratiu)
  A Model for the Retina Including Blood Flow and Deformation, A. Dziubek, E. Rusjan
• 11th USNCCM, Minneapolis, MN, July 28, 2011, Minisymposium Geometric and Exterior Calculus Methods in Computational Mechanics (Robert Haber, Douglas Arnold)
  Discrete Exterior Calculus for Biomedical Applications, A. Dziubek, G. Guidoboni
• Workshop on Moving Frames in Geometry, Centre de recherches mathematiques (CRM) Universite de Montreal, CA, June 17, 2011 (N. Kamran, A. Smith)
  Modeling Biomembranes using Calculus of Variations and Exterior Calculus
• Discrete Differential Geometry and Applications, 2010 Barrett Memorial Lectures, The Universiy of Tennessee, TN, May 17-21 2010 (Ken Stephenson, Alexander Bobenko)
  Poster: Discrete Differential Geometry for Closed Elastic Biomembranes

Applied Math Seminars, External Speakers, Events, SUNY Poly

• Wenfeng Chen (SUNYPoly, Mathematics), Introduction to Differential Operator Method for Solving Linear Nonhomogeneous Differential Equations with Constant Coefficients, September 9, 2016, noon, in DON 2159
• Souhayl Sadik, Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta A Geometric Theory of Nonlinear Morphoelastic Shells 15 April 2016, 2pm, DON 2159
• Kaushik Kalyanaraman, Computer Science and Applied Mathematics, SCoReC at Rensaler Polytechnic Institute, Troy, NY, Donuts, pretzels, fluid flow, and ranking sports teams: a short tour of discretizing Laplacians on meshes and graphs. 25 March 2106, 2pm, DON 2159
• Wenfeng Chen (SUNYPoly, Mathematics), An Attempt to discretize Witten's Deformation using concepts from Discrete Exterior Calculus, March 20, 2015, 2pm, in DON 2159
• Tanzid Sultan (SUNYPoly Math Major), Mercury's orbit around the Sun, March 5, 2015, noon, DON 2159
• Robert Lowry, SUNY Suffolk, Gauged Poisson Structures and Compressible Free Boundary Fluids, , April 25, 2014, noon, DON 2159
• Alexandra Benavente, SUNY College of Optometry, NYC, Modeling ocular blood flow to help prevent blindness, March 28, 2014, noon, Student Center Theater (Provost Lecture series)
• Michael Karow, University of Technology Berlin, Germany, Pseudospectra for structured matrix perturbations, May 30, 2013, 1pm, DON G152
• Holger Heumann, Visiting Scholar at Rutgers University, NJ, Lagrangian and Eulerian Methods for Generalized Advection-Diffusion, April 19, 2013, 12pm, DON G152
• Applied Mathematics Seminar
  • STEMUR Summer 2014: Ocular Blood Flow Modeling and Retinal Image Processing, Poster Presentations
    Saw Franco, Zach Harvey, James Huchital, Paul Lastowicka, Joe Pasqualetti, Daniel Yaciuk
    (Students from Applied Mathematics, Engineering, and Computer Science)
    29 August, 2014, DON 2159
  • Mathematical Modeling of retinal vascular structure, Multiscale Methods, Image Segmentation (Spring 2014)
  • SPNM & Image Processsing Tools for ODEs & PDEs with Applications in Medical Science, Data Analysis, and Engineering (Fall 2013)
  • Structure Preserving Numerical Methods for ODEs and PDEs (Spring 2013)
  • Geometry and Symmetry Based Numerical Methods (Fall 2012)
  • Discrete Exterior Calculus (Spring 2012)
  • Exterior Calculus and Discrete Exterior Calculus (Fall 2011)

Conferences/Workshops attended

• BIRS 5-day workshop: Geometric Mechanics Formulations for Continuum Mechanics, International Research Station for Mathematical Innovation and Discovery, Alberta, CA, 6/11-16, 3/16 - 3/21, 2025 (upcoming, invited)
• APS - Topical Symposium of the New York State Section of the American Physical Society Fall 2024: Advances in Computational Physics and the Physics of Computation, SUNY Polytechnic, Utica, 10/26, 2024
• AI+X - Second USF Artificial Intelligence + X Symposium, Institute for Artificial Intelligence, University of Florida, 10/24, 2024
• SIAM AN 24 - Society for Industrial and Applied Mathematics Annual Meeting, Sessions on Numerical Methods for Geometric PDEs, Spokane, Washington, 7/2024.
• MFO - Workshop 2225, Hilbert Complexes: Analysis, Applications, and Discretizations, Oberwolfach Research Institute for Mathematics, Black Forrest, Germany, 6/19-25, 2022 (invited)
• PCTS - Structure-Preserving Geometric Discretization of Physical Systems, Topics: Structure-preserving algorithms, Geometric integration, Geometric particle-in-cell methods, Symmetries in lattice quantum field theories, Princeton Center for Theoretical Sciences, Princeton University, NJ, 2/17-18, 2020
• INI - Geometry, compatibility and structure preservation in computational differential equations, GCSW02 - Structure preservation and general relativity, Isaac Newton Institute, Cambridge, UK, 9/30- 10/4, 2019
• SNP - 55th Meeting of the Society for Natural Philosophy: Microstructure, defects, and growth in mechanics, Loyola University Chicago, IL, 9/13-15, 2019
• BIRS 5-day workshop: Connections in Geometric Numerical Integration and Structure-Preserving Discretization, International Research Station for Mathematical Innovation and Discovery, Alberta, CA, 6/11-16, 2017
• 4th Biennial Int. Symposium on Age Macular Degeneration (AMD), Harward Medical School, Department of Opthalmology, Boston, 10/20-22, 2016, (P. A'Amore)
• CMS Winter Meeting, Session on on Recent Advances in Structure-Preserving Discretizations, Niagara Falls, Canada, December 2-5, 2016 (A. Bihlo, A. Wan)
• NYU-Oxford Workshop on Mathematical Models of Defects and Patterns, PIRE event, Courant Institute, NYU, NYC, 1/5-8, 2016 (R. V. Kohn, J. M. Ball, G. Leoni, P. Palffy-Muhoray)
• IGAschool2014, Isogeometric Analysis, Fundamentals and Applications, ECCAM, Barcelona, Spain, 7/27, 2014 (E.H. van Brummelen, T.J.R. Hughes, T. Kvamsdal, A. Reali, Sunday only)
• From the Clinic to Partial Differential Equations and Back: Emerging challenges for Cardiovascular Mathematics, ICERM, Brown University, Providence, RI, 1/20-24, 2014 (L Grinberg, AM Robertson, PJ Blanco, A Veneziani, JN Oshinski, WR Taylor)
• AMS Fall Central Sectional Meeting, University of Akron, Akron, OH, 10/20-21, 2012, Special Sessions on Applied Topology
• NSF/CBMS Conference: Finite Element Exterior Calculus (FEEC), ICERM, Brown University, Provience, RI, 6/11-15, 2012 (A. Demlow, J. Guzman, D. Leykekhman)
• DDG@SoCG (Geometry Processing and Physical Simulations), 28th ACM Symposium on Computational Geometry, Chapel Hill, NA, 6/18-20, 2012 (A. Hirani, M. Desbrun)

Seminars and Workshop, IUPUI

• Applied Math Seminar (Fall 2010)
• Workshop Discrete Differential Geometry for Multiphase Flow Problems (Workshop Webpage)
  April 2010: Organizer, with G. Guidoboni
• Red Blood Cell Seminar (Spring 2010)
• Lecture Series on Modelling Red Blood Cells using Differential Forms (Fall 2009)
  Text: Differential Forms with Applications to the Physical Sciences, Harley Flanders, Dover, 1989
  The Geometry of Physics, Theodore Frankel, Cambridge, 2004
• Lecture Series on Modelling Red Blood Cells using Shell Theory, Invited Speaker. (Summer 2008)
  Text: Cosserat Theories, Shells, Rods and Points, M.B. Rubin, Kluwer, 2000

Classes taught, SUNY Poly (SUNY IT), IUPUI, RHIT

• MAT 491 Independent Study, Finite Element Exterior Calculus and Mixed Methods for model problems, using NGSolve (Fall 2023)
• MAT 112 Elements of Calculus (Fall 2011, 2 sections. Spring 2012)
• MAT 151 Calculus I (Spring 2021)
• MAT 152 Calculus II (Fall 2013, Spring/Fall 2014)
• MAT 230 Ordinary Differential Equations (Fall 2019)
• MAT 253 Calculus III (2012 -- 2017, Fall 2020, 2023)
• MAT 260 Ordinary Differential Equations & Series Solutions
• MAT 280 Linear Algebra (Fall 2017, 2018, 2019, 2021, 2022, Spring 2020, 2022, 2023)
• MAT 423 Vector and Tensor Calculus (Fall 2016, 2021)
• MAT 460 Numerical Differential Equations (Springs 2015 -- 2023)
• MAT 480 Advanced Linear Algebra (undergraduate) (Spring/Fall 2021)
• MAT 502 Linear Algebra (graduate), Advanced Certificate CMM (Fall 2020)
• MAT 560 Numerical Differential Equations, Advanced Certificate CMM (Spring 2020)
• IUPUI: Vector Calculus, Multidimensional Mathematics, Multivariate Calculus, Analytic Geometry and Calculus II
• RHIT: Calculus II, Differential Equations, Matrix Algebra and Systems of Differential equations, Vector Calculus

Employment

• current: Associate Professor, Department of Mathematics and Physics, SUNY Polytechnic Institute (Former State University of New York Institute of Technology), NY
• 2011-2018: Assistant Professor, Department of Mathematics and Physics, SUNY Polytechnic Institute (Former State University of New York Institute of Technology), NY
• 2009-2011: Research Associate, Department of Mathematical Sciences, Indiana University Purdue University Indianapolis (IUPUI), IN
• 2007-2008: Visiting Assistant Professor, Department of Mathematics, Rose-Hulman Institute of Technology (RHIT), IN
• 2004-2006: Part Time Lecturer, Project Group Applied Mathematics, Institute of Mathematics, Berlin University of Technology (TUB), Germany
• 1998-2004: Co-chair of the Project Group Applied Mathematics, Institute of Mathematics, TUB, Germany