Prof. Krzysztof Wiśniewski, Ph.D., Dr. Habil., Eng. 

Doctoral thesis
1985  Analiza numeryczna statycznej stateczności powłoki cylindrycznej obciążonej wiatrem
 421 
Habilitation thesis
1998  Finite rotations of shells and beams. Extended equations and numerical models 
Professor
20110715  Title of professor 
Supervision of doctoral theses
1.  20180426  Jarzębski Paweł  Zastosowanie algorytmów wielowątkowych i rozproszonych do zwiększenia efektywności Metody Elementów Skończonych  1256  
2.  20081030  Panasz Przemysław  Nieliniowe modele powłok z 6 stopniami swobody bazujące na dwustopniowych aproksymacjach  617  
3.  20080626  Sadłowski Piotr  Parametryzacje rotacji i algorytmy rozwiązywania równań dynamiki z rotacyjnymi stopniami swobody  612 
Recent publications
1.  Meissner M., Wiśniewski K., Influence of room modes on lowfrequency transients: Theoretical modeling and numerical predictions, JOURNAL OF SOUND AND VIBRATION, ISSN: 0022460X, DOI: 10.1016/j.jsv.2019.02.012, Vol.448, pp.1933, 2019 Abstract: In the lowfrequency range, a sound reproduction in enclosures is strongly influenced by excited room modes. While the spectral impact of acoustic modes on a room response is well recognized, there is no sufficient knowledge on how these modes affects transients. In the paper this issue has been examined theoretically and numerically for a room excited by a tone burst by using a modal expansion method supported by a computer implementation. To quantify a temporal accuracy of a sound reproduction, the new metrics referred to as the tone burst reproduction error was introduced. The basis for determining this quantity was a deviation between the tone burst amplitude and the amplitude of a sound pressure computed via the Hilbert transform. A numerical simulation was performed for an irregularly shaped enclosure having a form of tworoom coupled system. Calculation results have proved that a high inaccuracy of a tone burst reproduction occurs at receiving points corresponding to sharp dips in a distribution of the steadystate sound pressure level. This is because in these points an amplitude of transient sound is much bigger than a tone burst amplitude. It was discovered that strong narrow peaks in the tone burst reproduction error are located at centers of vortices in the active sound intensity vector field. An influence of a sound damping in a room on a reproduction of a tone burst was also examined and it was found that a substantial increase in a wall sound absorption does not significantly improves a tone burst reproduction because it does not eliminate sharp dips in a distribution of the steadystate sound pressure level. Keywords:room acoustics, room modes, transients, tone burst, discrete Hilbert transform, sound intensity vector field Affiliations:
 
2.  Nowak Z., Nowak M., Pęcherski R.B., Wiśniewski K., Widłaszewski J., Kurp P.^{♦}, Computational modeling of thermoplastic behavior of inconel 718 in application to laserassisted bending of thinwalled tubes, INTERNATIONAL JOURNAL FOR MULTISCALE COMPUTATIONAL ENGINEERING, ISSN: 15431649, Vol.17, No.3, pp.317338, 2019 Abstract: Laserassisted tube bending is a promising manufacturing process which enables production of forms and shapes that cannot be obtained by purely mechanical bending. It is particularly suitable for high hardness and brittle materials, such as nickel alloys, ceramics and cast iron. In the current paper, mechanical loading and simultaneous heating by a moving laser beam are used in a controlled manner to obtain the required deformation. Experimental investigation of the Inconel 718 (IN718) alloy provides the basis for identification of parameters of two constitutive models, which encompass softening phenomena and the coupling of temperature and strains. Numerical simulations are conducted to provide more insight into the laserassisted bending process of the IN718 thinwalled tubes. Temperature, stress and deformation fields are determined in sequentially coupled thermomechanical analyses using the FE code ABAQUS. Laser beam is modeled as a surface heat flux using the dedicated DFLUX procedure. The temperature field is used as a thermal load in the static general step, together with an external mechanical load. The process of tube bending is controlled by the displacement of the piston rod of the actuator, while the thrust force is the resulting value. Keywords:laserassisted bending of tubes, identification of material parameters, numerical simulations Affiliations:
 
3.  Wiśniewski K., Turska E.^{♦}, Improved ninenode shell element MITC9i with reduced distortion sensitivity, COMPUTATIONAL MECHANICS, ISSN: 01787675, DOI: 10.1007/s0046601715104, Vol.62, No.3, pp.499523, 2018 Abstract: The 9node quadrilateral shell element MITC9i is developed for the ReissnerMindlin shell inematics, the extended potential energy and Green strain. The following features of its formulation ensure an improved behavior: 1. The MITC technique is used to avoid locking, and we propose improved ransformations for bending and transverse shear strains, which render that all patch tests are passed for the regular mesh, i.e. with straight element sides and middle positions of midside nodes and a central node. 2. To reduce shape distortion effects, the socalled corrected shape functions of Celia and Gray (Int J Numer Meth Eng 20:1447–1459, 1984) are extended to shells and used instead of the standard ones. In effect, all patch tests are passed additionally for shifts of the midside nodes along straight element sides and for arbitrary shifts of the central node. 3. Several extensions of the corrected shape functions are proposed to enable computations of nonflat shells. In particular, a criterion is put forward to determine the shift parameters associated with the central node for nonflat elements. Additionally, the method is presented to construct a parabolic side for a shifted midside node, which improves accuracy for symmetric curved edges. Drilling rotations are included by using the drilling Rotation Constraint equation, in a way consistent with the additive/multiplicative rotation update scheme for large rotations. We show that the corrected shape functions reduce the sensitivity of the solution to the regularization parameter γ of the penalty method for this constraint. The MITC9i shell element is subjected to a range of linear and nonlinear tests to show passing the patch tests, the absence of locking, very good accuracy and insensitivity to node shifts. It favorably compares to several other tested 9node elements. Keywords:9node shell element MITC9i, Twolevel approximation of strains, Patch tests, Corrected shape functions, Node shift parameters, Coarse mesh accuracy, Drilling rotations Affiliations:
 
4.  Jarzębski P., Wiśniewski K., Evaluation of Partial Factorization for Reduction of Finite Element Matrices, ENGINEERING TRANSACTIONS (ROZPRAWY INŻYNIERSKIE), ISSN: 0867888X, Vol.65, No.1, pp.163170, 2017 Abstract: In this paper, we present the concept of Partial Factorization [1] and discuss its possible applications to the Finite Element method. We consider: (1) reduction of the element tangent matrix, which is particularly important for mixed/enhanced elements and (2) reduction of the subdomain matrices of the Domain Decomposition (DD) equation solvers run either sequentially on a single machine or in parallel on a cluster of computers. We demonstrate that Partial Factorization can be beneficial for these applications. Keywords:multiscale models of multilayer shells, mixed/enhanced finite elements, parallel computing, domain decomposition, solvers Affiliations:
 
5.  Jarzębski P., Wiśniewski K., Taylor R.L.^{♦}, On parallelization of the loop over elements in FEAP, COMPUTATIONAL MECHANICS, ISSN: 01787675, DOI: 10.1007/s004660151156z, Vol.56, pp.7786, 2015 Abstract: In this paper, we consider parallelization of the loop over elements using OpenMP in FEAP (Taylor, 2014), which is a research FE code, very popular at universities. Even for a serial version of FEAP (a cluster version also exists) such a parallelization is a nontrivial task due to the existing architecture of this code, which complicates efficient parallelization. First, we compare the serial version of FEAP to the parallel code Warp3D (Dodds et al., 2014), considering the usage of time and memory. As we found, Warp3D is much faster but uses more memory than FEAP. An analysis of Warp3D helps us to devise our method of parallelization of the loop over elements. Next, we describe several changes in FEAP, which were necessary to parallelize the loop over elements using OpenMP. In particular, the subroutine assembling elemental matrices is identified as crucial to good performance, and several directives for the mutual exclusion synchronization of OpenMP are implemented and tested. Finally, we demonstrate the performance of the parallelized FEAP, designated as ompFEAP, on numerical examples involving 3D and shell elements of FEAP as well as user’s elements. We conclude that ompFEAP, using the directive ATOMIC for synchronization of the assembling, provides a very good speedup and efficiency. Keywords:Parallelization, OpenMP, Finite element method, FEAP, 3D and shell elements Affiliations:
 
6.  Wiśniewski K., Panasz P., Two improvements in formulation of ninenode element MITC9, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 00295981, DOI: 10.1002/nme.4399, Vol.93, pp.612634, 2013 Abstract: The paper concerns a wellknown twodimensional ninenode quadrilateral element MITC9, which is based on twolevel approximations of strains (assumed strain method). The element has good accuracy, but does not pass the patch test. ninenode element, twolevel approximation of strains, assumed strain method, twodimensional MITC9, patch test, sensitivity to shape distortions, coarse mesh accuracy Affiliations:
 
7.  Panasz P., Wiśniewski K., Turska E.^{♦}, Reduction of mesh distortion effects for ninenode elements using corrected shape functions, FINITE ELEMENTS IN ANALYSIS AND DESIGN, ISSN: 0168874X, DOI: 10.1016/j.finel.2012.11.003, Vol.66, pp.8395, 2013 Abstract: The paper concerns twodimensional ninenode quadrilateral elements based on the Green strain and the twolevel approximations of strains. These approximations reduce locking well for regular meshes but cannot prevent the drop of accuracy when the side and central nodes are shifted from the middle positions. Twodimensional ninenode elements, Corrected shape functions, Twolevel approximations of strains, Patch tests, Shape distortions Affiliations:
 
8.  Wiśniewski K., Turska E.^{♦}, Fournode mixed HuWashizu shell element with drilling rotation, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 00295981, DOI: 10.1002/nme.3335, Vol.90, pp.506536, 2012 Abstract: In this paper, enhanced fournode shell elements with six DOFs/node based on the Hu–Washizu (HW) functional are developed for Green strain. The drilling rotation is included through the drilling rotation constraint equation. The key features of the approach are as follows. fournode mixed shell element with six DOFs/node, pure or partial Hu–Washizu functionals, drilling rotation, optimal representations, skew coordinates Affiliations:
 
9.  Wiśniewski K., Wagner W.^{♦}, Turska E.^{♦}, Gruttmann F.^{♦}, Fournode HuWashizu elements based on skew coordinates and contravariant assumed strain, COMPUTERS AND STRUCTURES, ISSN: 00457949, DOI: 10.1016/j.compstruc.2010.07.008, Vol.88, pp.12781284, 2010 Abstract: Mixed 4node elements based on the Hu–Washizu (HW) functional are developed for the representation of the assumed strain in the natural basis at the element’s center, i.e. for the contravariant transformation rule. In other aspects, the formulation is identical as in our previous paper [9], to which this note is an addendum. Fournode finite elements, Hu–Washizu functional, Plane stress, Mixed elements, Skew coordinates, Contravariant assumed strain Affiliations:
 
10.  Wiśniewski K., Turska E.^{♦}, Improved fournode HuWashizu elements based on skew coordinates, COMPUTERS AND STRUCTURES, ISSN: 00457949, DOI: 10.1016/j.compstruc.2009.01.011, Vol.87, pp.407424, 2009 Abstract: Mixed 4node elements based on the Hu–Washizu (HW) functional are developed for stress and strain representations in various coordinates, including the skew, natural and Cartesian ones. The HW functional is used in incremental form, suitable for nonlinear materials. The key features of our approach are as follows. 4Node finite elements, Plane stress, Incremental Hu–Washizu functional, Mixed elements, Mixed/enhanced elements, Skew coordinates Affiliations:
 
11.  Wiśniewski K., Turska E.^{♦}, Improved fournode HellingerReissner elements based on skew coordinates, INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, ISSN: 00295981, DOI: 10.1002/nme.2343, Vol.76, pp.798836, 2008 Abstract: Mixed fournode elements based on the Hellinger–Reissner (HR) functional are developed for stress representations in various coordinates, including the skew, natural and Cartesian ones. The twofield HR functional is used in the classical form and in the incremental form suitable for nonlinear materials. fournode finite elements, incremental Hellinger–Reissner functional, assumed stress element, assumed stress/enhanced strain element, skew coordinates Affiliations:
 
12.  Panasz P., Wiśniewski K., Ninenode shell elements with 6 dofs/node based on twolevel approximations, FINITE ELEMENTS IN ANALYSIS AND DESIGN, ISSN: 0168874X, DOI: 10.1016/j.finel.2008.05.002, Vol.44, pp.784796, 2008 Abstract: The paper concerns 9node quadrilateral shell elements derived for Reissner's kinematics. They are based on the Green strain and potential energy, and are applicable to large (unrestricted) rotations. The characteristic features of the developed elements are as follows: Ninenode shell elements, 6 dofs/node, Drilling rotation, Twolevel approximation, Assumed strain, Selective reduced integration Affiliations:
 
13.  Wiśniewski K., Kowalczyk P., Turska E.^{♦}, Analytical DSA for explicit dynamics of elasticplastic shells, COMPUTATIONAL MECHANICS, ISSN: 01787675, DOI: 10.1007/s0046600600683, Vol.39, No.6, pp.761785, 2007 Abstract: The paper presents an analytical constitutive design sensitivity analysis (DSA) algorithm for explicit dynamics of elasticplastic finite rotation shells. Two explicit dynamical algorithms for finite rotation shells are presented, and the DSA is developed for the one formulated in terms of the rotation vector and its time derivatives, {ψ,ψ˙,ψ¨}. The hypoelastic constitutive model based on the GreenMcInnisNaghdi stress rate is used to derive an incremental algorithm in terms of ‘backrotated’ objects. The associative deviatoric HuberMises plasticity modified by plane stress conditions is implemented in the form suitable for finite rotation/small elastic strain increments. The analytical DSA is developed for the abovespecified problem, with the design derivatives calculated w.r.t. material parameters. Designdifferentiation of the dynamic algorithm and the scheme of handling the history data and the predicted values in differentiation, which is crucial in computing correct derivatives, are described. Besides, we show how to avoid Newton loops in the DSA algorithm, when such a loop is present in the constitutive algorithm. Numerical examples show that, despite a great complexity of the solution algorithm for the finiterotation elasticplastic shells, it is feasible to compute analytical design derivatives of very good accuracy. Keywords:Explicit dynamics, Finite rotation shell, Elasticplastic material, Analytical Design Sensitivity Analysis for constitutive parameters Affiliations:
 
14.  Wiśniewski K., Turska E.^{♦}, Enhanced Allman quadrilateral for finite drilling rotations, COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, ISSN: 00457825, DOI: 10.1016/j.cma.2005.11.003, Vol.195, pp.60866109, 2006 Abstract: The paper concerns a fournode quadrilateral element based on Allman shape functions undergoing finite (unrestricted) drilling rotations, and aims at improving its accuracy and facilitating its implementation. New Allman shape functions for finite drilling rotations, Enhanced Assumed Displacement Gradient method for formulations with rotations, Enhanced Allman finite elements Affiliations:
 
15.  Wiśniewski K., Finite Rotations of Shells and Beams Extended Equations and Numerical Models (Praca habilitacyjna), Prace IPPT  IFTR Reports, ISSN: 22993657, No.9, pp.1178, 1997  
16.  Wiśniewski K., Analiza numeryczna statycznej stateczności powłoki cylindrycznej obciążonej wiatrem (Praca doktorska), Prace IPPT  IFTR Reports, ISSN: 22993657, No.50, pp.1144, 1985 
List of recent monographs
1. 1  Wiśniewski K., Finite rotation shells: Basic equations and finite elements for Reissner kinematics, Lecture Notes on Numerical Methods in Engineering and Sciences, Springer Verlag, pp.1498, 2010 
List of chapters in recent monographs
1. 613  Wiśniewski K., Turska E.^{♦}, Recent Developments in the Theory of Shells, rozdział: On Performance of NineNode Quadrilateral Shell Elements 9EAS11 and MITC9i, Springer, pp.711725, 2019  
2. 507  Wiśniewski K., Turska E.^{♦}, Shelllike Structures. Advanced Theories and Applications, rozdział: Selected topics on mixed/enhanced fournode shell elements with drilling rotation, Springer International Publishing, 572, pp.247288, 2017  
3. 51  Wiśniewski K., Turska E.^{♦}, Shelllike Structures. Nonclassical Theories and Applications, rozdział: Recent Improvements in HuWashizu Shell Elements with Drilling Rotations, Springer, pp.391412, 2011 
Editor of monographs
1. 614  Altenbach H.^{♦}, Chróścielewski J.^{♦}, Eremeyev V. A.^{♦}, Wiśniewski K., Recent Developments in the Theory of Shells, Springer, pp.1798, 2019 
Conference papers
1.  Jarzębski P., Wiśniewski K., Performance of the parallel FEAP in calculations of effective material properties using RVE, CMM, 3rd Polish Congress of Mechanics and 21st International Conference on Computer Methods in Mechanics , 20150908/0911, Gdańsk (PL), pp.241244, 2016 Abstract: The paper concerns parallelization of an FE code for machines with shared memory in order to speed up computations of large models. parallelization, OpenMP, finite element method, FEAP, RVE, shells Affiliations:

Conference abstracts
1.  Wiśniewski K., Turska E.^{♦}, Recent improvements to ninenode shell element MITC9 with drilling rotations, SSTA 2017, Shell Structures: Theory and Applications, 20171011/1013, Gdańsk (PL), Vol.4, pp.399402, 2018 Abstract: The paper describes our improved 9node quadrilateral shell element MITC9i, which is derived for the ReissnerMindlin shell kinematics, the extended potential energy functional and Green strain. 9node shell element MITC9,twolevel approximation of strains, corrected shape functions, node shift parameters, drilling rotations Affiliations:
 
2.  Nowak Z., Nowak M., Pęcherski R., Wiśniewski K., Widłaszewski J., Kurp P.^{♦}, Computational Modelling of Thermoplastic Behaviour of Inconel 718 in Application to LaserAssisted Bending of ThinWalled Alloy Tubes, WCCM2018, 13th World Congress on Computational Mechanics, 20180722/0727, Nowy Jork (US), pp.1, 2018  
3.  Wiśniewski K., Turska E.^{♦}, Recent results on ninenode shell elements using twolevel approximation of strain, SolMech 2016, 40th Solid Mechanics Conference, 20160829/0902, Warszawa (PL), No.P122, pp.12, 2016 Keywords: finite element method, shell elements Affiliations:
 
4.  Jarzębski P., Wiśniewski K., Evaluation of partial factorization for condensation of shell and solidshell elemental matrices, SolMech 2016, 40th Solid Mechanics Conference, 20160829/0902, Warszawa (PL), No.P100, pp.12, 2016 Keywords: finite element methods, solidshell elements Affiliations:
 
5.  Jarzębski P., Wiśniewski K., On calculation of effective material properties using RVE method by parallelized FE code for shell applications, PCMCMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 20150908/0911, Gdańsk (PL), pp.375376, 2015 Abstract: This paper concerns parallelization of an FE code for machines with shared memory in order to speed up computations of large models. We parallelized the loop over elements in the research code FEAP using OpenMP, which required several modifications of the code and a specific method of synchronization for assembling, for details see [2]. The parallel solver was also applied. We demonstrate performance of the parallelized FEAP, designated as ’ompFEAP’, in calculations of effective properties of materials using the RVE method. Two RVE examples are computed, for a heterogenous metalceramic composite and for a ceramic foam with a complicated microstructure. We conclude that ompFEAP provides a very good speedup and efficiency causing only a small increase in memory usage. Keywords:parallelization, OpenMP, finite element method, FEAP, RVE, shells Affiliations:
 
6.  Jarzębski P., Wiśniewski K., Corrected shape functions for sixnode triangular element for heat conduction, CMM 2013, 20th International Conference on Computer Methods in Mechanics, 20130827/0831, Poznań (PL), pp.345353, 2014 Abstract: In this paper, we derived the corrected shape functions for 6node triangular element using the concept proposed in (Celia & Gray 1984). These shape functions were implemented in the twodimensional 6node triangular element for heat conduction as a replacement of the isoparametric ones. The numerical tests indicate that, for distorted meshes, the new element is more accurate than the standard element. Comparisons of the accuracy of a range of triangular and quadrilateral elements also are provided. Keywords:Sixnode triangular element, heat conduction, corrected shape functions, sensitivity to mesh distortion, patch test Affiliations:
 
7.  Jarzębski P., Wiśniewski K., On parallelization of the loop over elements for composite shell computations, SolMech 2014, 39th Solid Mechanics Conference, 20140901/0905, Zakopane (PL), pp.227228, 2014  
8.  Wiśniewski K., Turska E.^{♦}, Recent improvements in mixed/enhanced shell elements with drilling rotation, SolMech 2014, 39th Solid Mechanics Conference, 20140901/0905, Zakopane (PL), pp.2728, 2014  
9.  Wiśniewski K., Turska E.^{♦}, On mixed/enhanced HuWashizu shell elements with drilling rotation, SSTA, 10th Jubilee Conference on Shell Structures  Theory and Applications, 20131016/1018, Gdańsk (PL), DOI: 10.1201/b15684117, Vol.3, pp.469472, 2014 Abstract: Mixed/enhanced fournode shell elements with six dofs/node based on the HuWashizu (HW) functional are developed for Green strain. The shell HW functional is derived from the shell potential energy functional instead of from the threedimensional HW functional. Partial HW functionals, differing in the bending/twisting part and the transverse shear part, are obtained. For the membrane part of HW shell elements, a 7parameter stress, a 9parameter strain and a 2parameter EADG enhancement are selected as performing best. The assumed representations of stress and strain are defined in skew coordinates in the natural basis at the element's center. The drilling rotation is included through the drilling Rotation Constraint (RC) equation and the Perturbed Lagrange method. The spurious mode is stabilized using the gamma method. Several versions of shell HW elements are tested using several benchmark examples and the optimally performing element is selected (HW29) in (Wisniewski & Turska 2012). Affiliations:
 
10.  Jarzębski P., Wiśniewski K., On corrected shape functions for sixnode triangular elements applied to heat conduction problems, CMM 2013, 20th International Conference on Computer Methods in Mechanics, 20130827/0831, Poznań (PL), No.MS09, pp.1112, 2013  
11.  Wiśniewski K., Turska E.^{♦}, On Shell Elements Derived from HuWashizu Functional, SolMech 2012, 38th Solid Mechanics Conference, 20120827/0831, Warszawa (PL), pp.228229, 2012  
12.  Panasz P., Wiśniewski K., Modeling of intersections using the ninenode assumed strain shell element, CMM 2011, 19th International Conference on Computer Methods in Mechanics, 20110509/0512, Warszawa (PL), pp.16212, 2011  
13.  Panasz P., Wiśniewski K., On computation of sensitivities of multilayer shells using elements with additional parameters, CMM 2011, 19th International Conference on Computer Methods in Mechanics, 20110509/0512, Warszawa (PL), pp.27512, 2011  
14.  Panasz P., Wiśniewski K., On behavior of nonlinear ninenode shell elements in thin limit, SolMech 2010, 37th Solid Mechanics Conference, 20100906/0910, Warszawa (PL), pp.320321, 2010  
15.  Wiśniewski K., Kowalczyk P., Turska E.^{♦}, DSA for Elasticplastic Shells and Explicit Dynamics, 8th U.S. National Congress on Computational Mechanics, 20050724/0728, Austin, Texas (US), No.1681, pp.1, 2005 Keywords: design sensitivity analysis, finite element method, shell structures, elastoplasticity Affiliations:
 
16.  Wiśniewski K., Kowalczyk P., Turska E.^{♦}, DSA for elasticplastic finite rotation shells under dynamic loads, ICTAM XXI, 21st International Congress of Theoretical and Applied Mechanics, 20040815/0821, Warszawa (PL), No.12679, pp.361, 2004 Keywords: Design sensitivity analysis, finite element method, finite rotations, shell elements Affiliations:
 
17.  Wiśniewski K., Kowalczyk P., Turska E.^{♦}, DSA for elasticplastic finite rotation shells under dynamic loads, ICTAM04, 21st International Congress of Theoretical and Applied Mechanics, 20040815/0821, Warszawa (PL), No.12679, pp.12, 2004 Abstract: The paper describes a constitutive algorithm for elasticplastic finite rotation shells and explicit dynamics with design derivatives calculated w.r.t. We show that despite a great complexity of the solution algorithm for the finiterotation elasticplastic shells, it is feasible to compute analytical design derivative of this algorithm, and the yielded sensitivities are of very good accuracy. Keywords:design sensitivity analysis, finite elment method, shell structures, dynamics, finite rotations Affiliations:
