Calculus of variations and numerical techniques in solid mechanics

PN-III-P1-1.1-TE-2019-0348, Contract No. TE 8/2020

Duration: September 2020- August 2022

Financially supported by Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI

Host Institution: "Alexandru Ioan Cuza" University of Iasi


Principal Investigator: Dr. Ionel-Dumitrel Ghiba

Research Team:

• Prof. dr. Liviu Marin (Senior researcher)


• Conf. dr. Ionut Munteanu (Young researcher)


• ACS. drd. Bucataru Mihai (Ph.D. student)

Project Summary:

In this project, we shall focus on some recent and important mathematical problems in the framework of solid mechanics. The project requires a high degree of analytical and numerical mathematical sophistication, technical abilities and drive if it is to be moved forward across a wide range of issues: Characterization of convexity concepts in nonlinear elasticity, Calculus of variations in nonlinear Cosserat elastic shells models, Inverse problems in solid mechanics and Stabilization for models in solid mechanics.


Objectives:

• Characterization of convexity concepts in nonlinear elasticity.;


• Calculus of variations in nonlinear Cosserat shells model;


• Inverse problems in solid mechanics;


• Stabilization for models in solid mechanics.

Expected results:

We shall promote international scientific cooperation and we shall increase the visibility of Romanian research by publishing highly original research papers (a minimum of 7 articles in international ISI journals).

Delivered results: -

17 articles have been developed: 16 articles are published in ISI indexed journals, while one article is in the review process at an ISI indexed journal. The results have been disseminated at 16 conferences in the field, held by each of the 4 grant members. The objectives set were fully achieved, according to the plan and methodology set out in the contract.

Impact of the obtained results:

The obtained results have a major impact on the addressed research fields, but also in related areas, as all results have a strong applicative character. The methods used and the way the results are presented are innovative. The applications based on the results obtained in this grant are in the field of engineering/design of thin bodies, elastic, vascoelastic and thermoelastic media as well as microstructure media.

Highlighted result:

J. Voss, I.D. Ghiba, R.J. Martin, P. Neff. A rank-one convex, non-polyconvex isotropic function on $GL^+(2)$ with compact connected sublevel sets. Proceedings A of the Royal Society of Edinburgh, 152(2): 356-381, 2022.

Short description of the delivered results:

New mathematical models for microstructured elastic shells have been obtained and studied. The study of elastic shells is extremely important in various industrial processes, such as the aeronautical industry and the production of nanobodies. The models constructed are mathematically rigorous but, at the same time, are presented in a language accessible to a wide audience. It has been identified how the initial curvature of the initial configuration influences its subsequent deformation. Conditions on the thickness of the shells have also been deduced for which the proposed mathematical models lead to solutions of variational problems. It is worth mentioning that the proposed models are the first models in the literature capable of capturing some of the effects that have been observed experimentally.

Since many mechanical processes do not fit into the theory of small deformations (linear theory), mathematical modeling of nonlinear elastic processes is of great interest. In 2D nonlinear theory we have made an in-depth study of the properties of the energies of two classes widely used in practice: isochoric energies and energies in which the isochoric part and the volumetric part are additively coupled. New criteria have been identified to demonstrate rank-one convexity of such energies. For isochoric energies the exact form of the quasi-convex envelope has been identified, a calculation which is often impossible to perform. Also, alternative or first answers to some conjectures in the nonlinear elasticity of 2D have been given.

Inverse problems in solid mechanics represent a classical and mathematically very interesting type of problems which are very often encountered in numerous real life engineering problems (advanced machinery, biomedical applications, heat transfer, turbomachinery, etc.). Very accurate, convergent and stable methods for the reconstruction of various types (regular and non-smooth) of thermal fields on an inaccessible boundary and in the domain occupied by an anisotropic solid body from additional measurements taken on the accessible boundary are developed in the present grant, whilst an accurate, convergent and stable method for the recovery of a time-dependent heat source in isotropic thermos-elasticity systems of type-III is also provided. The aforementioned numerical methods developed, which are based on appropriate variational formulations of the problems, open new avenues for the approach of various linear and non-linear inverse problems associated with solid mechanics.

Regarding the last objective, the obtained results have an important impact on the problem of non-symmetric equations characterizing physical (mechanical) phenomena, which is a highly difficult subject. We designed stabilizing feedback controllers which are with delay, which means that, when used in practice, they allow for time to pass until their action, which gives time to do the necessary measurements. Other important results study the problem of constructing stabilizing controllers for equations modelling physical phenomena for which the initial data is not known. In this case, the initial data is replaced by an weighted mean of repeated measurements. We also studied the problem of stabilization for the equations where we added random perturbations, attempting to make the model closer to the real physical phenomena. Other results concern the problem of stabilization of binary-mixture materials.

Publications:

• I.D. Ghiba, M. Birsan, P. Lewintan, P. Neff. The isotropic Cosserat shell model including terms up to $O(h^5)$. Part I: Derivation in matrix notation, Journal of Elasticity, 142:201-262, 2020.


• R.J. Martin, J. Voss, I.D. Ghiba, O. Sander, P. Neff. The quasiconvex envelope of conformally invariant planar energy functions in isotropic hyperelasticity, Journal of Nonlinear Science, 30:2885–2923, 2020.


• I.D. Ghiba, P. Neff, S. Owczarek. Existence results for non-homogeneous boundary conditions in the relaxed micromorphic model, Mathematical Methods in the Applied Sciences, 44:2040-2049, 2021.


• I. Munteanu. Exponential stabilization of the semilinear heat equation with nonlocal boundary conditions, Journal of Mathematical Analysis and Applications, 492 (2):124512, doi.org/10.1016/j.jmaa.2020.124512, 2020.


• J. Voss, I.D. Ghiba, R.J. Martin, P. Neff. Sharp rank-one convexity conditions in planar split, isotropic elasticity for the additive volumetric-isochoric. Journal of Elasticity, 143, 301-335, 2021.


• J. Voss, I.D. Ghiba, R.J. Martin, P. Neff. A rank-one convex, non-polyconvex isotropic function on $GL^+(2)$ with compact connected sublevel sets. Proceedings A of the Royal Society of Edinburgh, 152(2): 356-381, 2022.


• I.D. Ghiba, M. Birsan, P. Lewintan, P. Neff. A constrained Cosserat shell model up to order $O(h^5)$ : Modelling, existence of minimizers, relations to classical shell models and scaling invariance of of the bending tensor. Journal of Elasticity, 146 (1): 83-141, 2021.


• G. Rizzi, G. Hutter, H. Khan, I.D. Ghiba, A. Madeo, P. Neff. Analytical solution of the cylindrical torsion problem for the relaxed micromorphic continuum and other generalized continua (including full derivations). Mathematics and Mechanics of Solids, 27:507-553, 2022.


• S. Owczarek, I.D. Ghiba, P. Neff. A note on local higher regularity in the dynamic linear relaxed micromorphic model. Mathematical Methods in the Applied Sciences, 44:13855-13865, 2021.


• M. Bucataru, I. Cimpean, L. Marin. A gradient-based regularization algorithm for the Cauchy problem in steady-state anisotropic heat conduction, Computers & Mathematics with Applications, 119:220-240, 2022.


• K. Van Bockstal, L. Marin. The reconstruction of a time-dependent heat source in isotropic thermoelasticity systems of type-III, ZAMP -- Zeitschrift fur Angewandte Mathematik und Physik, 73(3): art. no. 113, 2022.


• M. Bucataru, L. Marin. FDM-based alternating iterative algorithms for inverse BVPs in 2D steady-state anisotropic heat conduction with heat sources, {Applied Mathematics and Computation} 2022, trimis spre publicare.


• I. Munteanu. Boundary stabilizing actuators for multi-phase fluids in a channel. Journal of Differential Equations, 285: 175-210, 2021.


• I. Munteanu. Boundary stabilization of non-diagonal systems by proportional feedback forms. Communications on Pure and Applied Analysis, 20(9): 3095-3110, 2021.


• D. Goreac, I. Munteanu. Improved stability for linear SPDEs using mixed boundary/internal controls. Systems & Control Letters 156:105024 2021.


• I. Munteanu. Feedback exponential stabilization of the semilinear heat equation with nonlocal initial conditions, Nonlinear Analysis: Modelling and Control, 26:1106-1122, 2021.


• I. Munteanu. Stabilisation of non-diagonal infinite-dimensional systems with delay boundary control, International Journal of Control, doi.org/10.1080/00207179.2022.2063193, 2022.

Conferences:

• L. Marin, M. Bucataru, I. Cimpean, An iterative algorithm for the Cauchy problems associated with the steady-state anisotropic heat conduction, Current Trends in Applied Mathematics – Online Workshop, “Octav Mayer” Institute of Mathematics, Romanian Academy, Iasi Branch, Iasi, Romania, 21-22 septembrie 2020.


• M. Bucataru, L. Marin, An iterative algorithm for the Cauchy problem associated with the steady-state anisotropic heat conduction. Conferinta Scolilor Doctorale din Consortiul Universitaria (CSDCU-MIF2020), Iasi, Romania, 22 octombrie 2020.


• I.D. Ghiba, Un nou model matematic pentru panze elastice, Sesiunea de comunicări stiintifice a Institutului de matematica „Octav Mayer” si a comisiei de automatica teoretica si teoria controlului optimal, Zoom Meeting, 17.10.2020.


• I.D. Ghiba, Despre un nou model in teoria panzelor elastice, Sesiunea de comunicări științifice a Facultății de Matematică, Webex Meeting, 30.10.2020.


• M. Bucataru, I. Cimpean, L. Marin, An iterative algorithm for the Cauchy problem in steady-state anisotropic heat conduction. Conferinta scolilor doctorale din Consortiul Universitaria pe domeniile Matematica, Informatica, Fizica (CSDCU--MIF~2020), Universitatea "Alexandru Ioan Cuza" din Iasi, 22--24 septembrie 2020, online.


• M. Bucataru, I. Cimpean, L. Marin, A GDM-based regularization algorithm for Cauchy inverse problems in anisotropic heat conduction. {Sesiunea de comunicari stiintifice pentru studenti}, Universitatea din Bucuresti, Facultatea de Matematica si Informatica, 8 mai 2021, online.


• M. Bucataru, I. Cimpean, L. Marin, A GDM-based regularization algorithm for Cauchy inverse problems in anisotropic heat conduction. Tenth Workshop for Young Researchers in Romania (WYRM~10), Institutul de Matematica "Simion Stoilow" al Academiei Romane, Institutul de Matematica "Octav Mayer" al Academiei Romane si Facultatea de Matematica si Informatica, Universitatea "Ovidius" din Constanta, 20--21 mai 2021, online.


• I.D. Ghiba, A new Cosserat shell model: modelling and existence of the solution, 20-21 mai 2021, Tenth Workshop for Young Researchers in Romania (WYRM~10), Institutul de Matematica "Simion Stoilow" al Academiei Romane, Institutul de Matematica "Octav Mayer" al Academiei Romane si Facultatea de Matematica si Informatica, Universitatea "Ovidius" din Constanta, 20--21 mai 2021, online.


• I.D. Ghiba, Modelling and existence results in the Cosserat shell theory, The 28th Conference on Applied and Industrial Mathematics, CAIM 2021, 17-18 September, online, 2021.


• I.D. Ghiba, Existenta si unicitate pentru problema propagarii undelor seismice in medii cu microstructura, Sesiunea de comunicari stiintifice a Institutului de Matematica Octav Mayer impreuna cu Comisia de Automatica teoretica si Teoria controlului, 30.10.2021, online.


• I.D. Ghiba, Propagarea undelor seismice în medii Cosserat, Sesiunea de comunicari stiintifice a Facultatii de Matematica in cadrul evenimentului Zilele Universitatii Alexandru Ioan Cuza din Iasi, 29.10.2021, online.


• I. Munteanu, Backstepping vs Direct-Proportional control design techniques, Sesiunea de comunicari stiintifice a Institutului de Matematica Octav Mayer impreuna cu Comisia de Automatica teoretica si Teoria controlului, 30.10.2021, online.


• M. Bucataru, I. Cimpean, L. Marin, Stable numerical reconstruction of non-smooth boundary data in steady-state anisotropic heat conduction. International Conference on Inverse Problems in Engineering (ICIPE~2022), Francavilla al Mare (Chieti), Italia, 15-19 mai 2022.


• M. Bucataru, I. Cimpean, L. Marin, Stable numerical reconstruction of non-smooth boundary data in steady-state anisotropic heat conduction. Sesiunea de comunicari stiintifice pentru studenti, Universitatea din Bucuresti, Facultatea de Matematica si Informatica, 27 mai 2022.


• M. Bucataru, I. Cimpean, L. Marin, Stable numerical reconstruction of non-smooth boundary data in steady-state anisotropic heat conduction. 8th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS~2022), Oslo, Norvegia, 5-9 iunie 2022.


• I. Munteanu, Boundary stabilization of parabolic type equations by proportional type feedback form, Nonlinear Problems of Mathematical Physics, Koc University, Ankara, Turcia, 14 ianuarie 2022.


• I. Munteanu, Controlled Stochastic PDEs, Stochastic Analysis Seminar, Imperial College, London, UK, 15 februarie 2022.

Academic visits:

• I. Munteanu, INSA-Rouen, 16.08.2021-20.09.2021, France.


• I. Munteanu, Universitat Bielefeld, 29.06.2022-31.07.2022, Germania.

Scientific reports:

• Scientific report 2020


• Scientific report 2021


• Scientific report 2022-Final report