dr inż. Maciej Zawidzki 

Doktorat
201009  Application of computational intelligence to engineering design problems in architecture firmitatis, utilitatis, venustatis (RUKJ)
 1332 
Ostatnie publikacje
1.  Zawidzki M., Szklarski J., Effective Multiobjective Discrete Optimization of TrussZ Layouts Using a GPU, APPLIED SOFT COMPUTING, ISSN: 15684946, DOI: 10.1016/j.asoc.2018.05.042, pp.122, 2018 Streszczenie: TrussZ (TZ) is an Extremely Modular System for creating skeletal freeform ramps and ramp networks. The TZ structures are comprised of four variations of two types of basic unit subjected to rotation. The two types of units are: R and L being a mirror reflection of each other. This paper presents a novel method based on image processing, evolutionary algorithm and intensive parallelization of multiobjective optimization of TZ layouts. Słowa kluczowe: TrussZ, Extremely Modular System, Retrofitting, accessibility, multiobjective, discrete, combinatorial, optimization, genetic algorithm, parallel computing, GPU, GPGPU Afiliacje autorów:
 40p.  
2.  Zawidzki M., Szklarski J., Transformations of ArmZ modular manipulator with Particle Swarm Optimization, Advances in Engineering Software, ISSN: 09659978, DOI: 10.1016/j.advengsoft.2018.05.003, pp.114, 2018 Streszczenie: A novel concept of hyperredundant, snakelike manipulator is presented. It is based on the reconfigurable modular construction system–ArmZ (AZ). AZ is comprised of linearly joined congruent modules with possibility of relative twist. AZ is an Extremely Modular System, i.e. it is composed of a single basic unit and allows for creating freeform shapes. Required level of usefulness and efficiency are among the most challenging design aspects of such reconfigurable systems. Here AZ is considered in the context of kinematics of robotic arms. In general, due to its highly nonlinear nature, it is very difficult to find transitions between given states (configurations), especially under realistic environmental and structural constraints. As a way to control the manipulator, an implementation of Particle Swarm Optimization (PSO) for finding transitions between AZ states in realistic scenarios is proposed. Four practical examples are presented which are variations of two distinct problems: bending of a hexagonal AZ in a narrow slot (strong environmental constraints), and reaching a given point in 3D space by the tip of dodecagonal AZ (acting as a robotic arm). The problem of AZ transformation has been defined as a multiobjective optimization. The methodology is general with no restrictions to the objective function. Since the problem is strongly nonlinear, in order to cover large space of potential solutions, the algorithm runs for a relatively large number of random initial swarms. This task was distributed on a computer cluster. Although the nature of AZ reconfiguration is discrete, the optimization algorithm is continuous. Słowa kluczowe: Extremely modular system, ArmZ, PipeZ, Discrete optimization, Dihedral rotation, Reconfiguration, Particle Swarm Optimization, Redundant robot, Hyperredundant manipulator Afiliacje autorów:
 30p.  
3.  Zawidzki M., Jankowski Ł., Optimization of modular TrussZ by minimummass design under equivalent stress constraint, SMART STRUCTURES AND SYSTEMS, ISSN: 17381584, DOI: 10.12989/sss.2018.21.6.715, Vol.21, No.6, pp.715725, 2018 Streszczenie: TrussZ (TZ) is an Extremely Modular System (EMS). Such systems allow for creation of structurally sound freeform structures, are comprised of as few types of modules as possible, and are not constrained by a regular tessellation of space. Their objective is to create spatial structures in given environments connecting given terminals without selfintersections and obstacleintersections. TZ is a skeletal modular system for creating freeform pedestrian ramps and ramp networks. The previous research on TZ focused on global discrete geometric optimization of the spatial configuration of modules. This paper reports on the first attempts at structural optimization of the module for a singlebranch TZ. The internal topology and the sizing of module beams are subject to optimization. An important challenge is that the module is to be universal: it must be designed for the worst case scenario, as defined by the module position within a TZ branch and the geometric configuration of the branch itself. There are four variations of each module, and the number of unique TZ configurations grows exponentially with the branch length. The aim is to obtain minimummass modules with the von Mises equivalent stress constrained under certain design load. The resulting modules are further evaluated also in terms of the typical structural criterion of compliance. Słowa kluczowe: Extremely Modular System, TrussZ, structural optimization, modular structures, minimum mass design, frame structures Afiliacje autorów:
 30p.  
4.  Zawidzki M.^{♦}, Deployable PipeZ, Acta Astronautica, ISSN: 00945765, DOI: 10.1016/j.actaastro.2016.05.023, Vol.127, pp.2030, 2016 Streszczenie: This paper presents a concept of deployable PipeZ (dPZ): a modular structural system which takes advantage of the robustness of rigidpanel mechanism and allows to create freeform links which are also reconfigurable and deployable. The concept presented can be applied for building habitats and infrastructures for human exploration of oceans and outer space. dPZ structures can adapt to changing requirements e.g. mission objectives, crew condition and technological developments. Furthermore, such lightweight and adaptable structural concept can assist in sustainable exploration development. After brief introduction, the concept of PipeZ (PZ) is presented. Next, the reconfigurability of PZ is explained and illustrated with continuous and collisionfree transition from a PZ forming a Trefoil knot to a Figureeight knot. The following sections introduce, explain and illustrate the folding mechanism of a single foldable PipeZ module (fPZM) and entire dPZ structure. The latter is illustrated with asynchronous (delayed) unfolding of a relatively complex Unknot. Several applications of PZ are suggested, namely for underwater and deepspace and surface habitats, for permanent, but in particular, temporary or emergency passages. As an example, a scenario of a failure of one of the modules of the International Space Station is presented where a rigid structure of 40 fPZMs bypasses the “dead link”. A lowfidelity prototype of a 6module octagonal dPZ is presented; several folding schemes including concentric toric rings are demonstrated. Practical issues of pressurization and packing are briefly discussed. Słowa kluczowe: Ocean and space outpost; Bananasplit; Deployable structure; Rigidpanel folding; Freeform Afiliacje autorów:
 35p.  
5.  Zawidzki M.^{♦}, Optimization of Multibranch TrussZ based on Evolution Strategy, Advances in Engineering Software, ISSN: 09659978, DOI: 10.1016/j.advengsoft.2016.07.015, Vol.100, pp.113125, 2016 Streszczenie: This paper concerns multibranch TrussZ networks (MTZ). A possible scenario for creating a “multibranch bridge” linking 6 terminals of pedestrian and cycling communication is presented. This process is formulated as a constrained minimization problem. New, biologyinspired nomenclature for MTZ and encoding for MTZ are introduced. Several operations for MTZs are introduced and illustrated. The functionality of these operations is illustrated with transformation from a random MTZ to a “proper” 6branch MTZ network. A populationbased heuristic experiment is presented to demonstrate that the introduced operators allow us to create any desirable MTZ. A cost function for the considered scenario is introduced. The genetic operations are interpreted and visualized. A number of feasible MTZ layouts produced by an evolution strategybased algorithm are presented. One of these layouts is used for creation of the spatial 6terminal MTZ, which is also visualized. Słowa kluczowe: Extremely modular system, Modular ramp system, Multibranch network, Modular structure encoding, Evolution strategy, Discrete layout optimization Afiliacje autorów:
 30p.  
6.  Zawidzki M.^{♦}, Automated geometrical evaluation of a plaza (town square), Advances in Engineering Software, ISSN: 09659978, DOI: 10.1016/j.advengsoft.2016.01.018, Vol.96, pp.5869, 2016 Streszczenie: This paper presents a method for an automated geometrical evaluation (AGE) intended as a design support tool for urban design of a plaza (P). AGE is based on three normalized properties derived from a plan of P, namely: smallness, enclosure, and regularity. 19 worldwide plazas have been evaluated by 20 respondents in, what is called here, human subjective evaluation (HSE). A brief analysis of HSE including the identification of redundant categories is presented. Two P evaluation methods based on all four (S,C,E,R) and selected three (S,E,R) properties are discussed. Good agreement of AGE based on S,E, and R (NPSER) with HSE is shown. P quality rating (excellent, good, fair) based on NPSER is introduced. Exceptional cases are briefly discussed Słowa kluczowe: Urban composition; Public square; Plaza; Layout evaluation; Design support tool; Normalized accumulated quality Afiliacje autorów:
 30p.  
7.  Zawidzki M.^{♦}, Dynamic shading of a building envelope based on rotating polarized film system controlled by onedimensional cellular automata in regular tessellations (triangular, square and hexagonal), Advanced Engineering Informatics, ISSN: 14740346, DOI: 10.1016/j.aei.2014.09.008, Vol.29, No.1, pp.87100, 2015 Streszczenie: The original prototype of the cellular automaton (CA) shading system (CASS) for building facades was based on rectangular array of cells and used liquid crystal technology. This paper introduces polarized film shading system (PFSS) – an alternative approach based on optomechanical modules whose opacity is a function of the rotation of polarized film elements. PFSS in regular tessellations: triangular, square and hexagonal are discussed. Simulations for each type of tessellation are presented and visualized. Visual attractiveness of emergent CA patterns manifested by ‘‘particles’’ and ‘‘solitons’’ is discussed. Słowa kluczowe: Adaptive architecture, Organic architecture, Building envelope, CA shading Afiliacje autorów:
 35p.  
8.  Zawidzki M.^{♦}, Retrofitting of pedestrian overpass by TrussZ modular systems using graphtheory approach, Advances in Engineering Software, ISSN: 09659978, DOI: 10.1016/j.advengsoft.2014.11.004, Vol.81, pp.4149, 2015 Streszczenie: Installing pedestrian ramps is a common improvement towards a barrierfree environment. This paper introduces a graphtheoretical method of retrofitting of a singlebranch TrussZ (TZ) ramp in a constrained environment. The results produced by this exhaustive search method are usually ideal and better than those produced previously with metaheuristic methods. A large case study of linking two sections of the Hongo Campus of Tokyo University using an overpass in an extremely constrained environment is presented. TZ modules with 1:12 (8.3%) slope are used, which is allowable in most countries for ramps for selfpowered wheelchairs. The results presented here are highly satisfactory both in terms of structural optimization and aesthetics. Visualizations of the TZ ramp system, composed of 124 units, are presented. Słowa kluczowe: TrussZ, Modular lightweight system, Organic design, Discrete structural optimization, Retrofitting, Pedestrian ramp, Breadthfirst search, Wavefront algorithm, Unknown graph exploration Afiliacje autorów:
 30p.  
9.  Zawidzki M.^{♦}, Chraibi M.^{♦}, Nishinari K.^{♦}, CrowdZ: The userfriendly framework for crowd simulation on an architectural floor plan, Pattern Recognition Letters, ISSN: 01678655, DOI: 10.1016/j.patrec.2013.10.025, Vol.44, pp.8897, 2014 Streszczenie: This paper introduces CrowdZ (CZ): a framework that provides a userfriendly platform where architects can perform simple crowd simulations on floor plans. A simple but robust and flexible agentbased system is used for modeling of the crowd dynamics. Such simulations can be performed at any stage of design – from rough sketches to the final blueprints. CZ allows acquiring the layouts for the simulations in a number of ways: freehand sketches, importing already prepared images and appropriating preprocessed images from commercially available Computer Aided Design programs. These three methods are illustrated with practical examples, followed by a number of simulations compared with the literature or other commercially available programs. Słowa kluczowe: Pedestrian dynamics; Agent based modeling; Design support; Digitized floor plan Afiliacje autorów:
 25p.  
10.  Zawidzki M.^{♦}, Nishinari K.^{♦}, Application of evolutionary algorithms for optimum layout of TrussZ linkage in an environment with obstacle, Advances in Engineering Software, ISSN: 09659978, DOI: 10.1016/j.advengsoft.2013.04.022, Vol.65, pp.4359, 2013 Streszczenie: TrussZ (TZ) is a concept of a modular system for creating freeform links and ramp networks. It is intended as a universal transportation system for cyclists and pedestrians, especially ones with strollers or carts, and in particular – by persons on wheelchairs, the elders, etc. In other words, TZ is for people who have difficulties using regular stairs or escalators. With only two types of modules, TZ can be designed for nearly any situation and therefore is particularity suited for retrofitting to improve the mobility, comfort and safety of the users. This paper presents an application of evolution strategy (ES) and genetic algorithm (GA) for optimization of the planar layout of a TZ linkage connecting two terminals in a given environment. The elements of the environment, called obstacles, constrain the possible locations of the TZ modules. Criteria of this multiobjective optimization are: the number of modules to be the smallest, which can be regarded as quantitative economical optimization, and the condition that none of the modules collides with any other objects, which can be regarded as qualitative satisfaction of the geometrical constraints. Since TZ is modular, the optimization of its layout is discrete and therefore has combinatorial characteristic. Encoding of a planar TZ path, selection method, objective (cost) function and genetic operations are introduced. A number of trials have been performed; the results generated by ES and GA are compared and evaluated against backtrackingbased algorithm and random search. The convergence of solutions is discussed and interpreted. A visualization of a realistic implementation of the best solution is presented. Further evaluation of the method on three other representative layouts is presented and the results are briefly discussed. Słowa kluczowe: TrussZ, Modular skeletal system, Organic design, Metaheuristic discrete optimization, Retrofitting, Pedestrian ramp Afiliacje autorów:
 25p.  
11.  Zawidzki M.^{♦}, Bator M.^{♦}, Application of Evolutionary Algorithm for Optimization of the Sequence of Initial Conditions for the Cellular AutomatonBased Shading, Journal of Cellular Automata, ISSN: 15575969, Vol.7, pp.363384, 2013 Streszczenie: This paper presents an application of evolutionary algorithm (EA) for multiobjective optimization of the sequence of initial conditions (SIC) for a cellular automaton (CA) used for a potential implementation in the field of architecture. In the proposed application, a modular shading system for building facade is driven by a two color, one dimensional, range 2 CA rule {3818817080,2,2}. The SIC optimization criteria are: visual attractiveness, gradual and intuitive transition from one density level to another and even distribution of the pattern over the entire array. The ideal solutions for 10 square arrays of 7×7, 8×8,..., 16×16 cells are found by an exhaustive search method – the backtracking. The encoding of SICs using the orderbased representation is introduced. A cost function evaluating both monotonicity of the average density transition, and the distribution of shading pattern is introduced. For a 100×100 cell array EA is implemented with three setups: without crossover but with intensive mutation, with crossover and without mutation, and with both crossover and mutation. Two types of crossover operations are used: uniform (UX) and onepoint (OPX). A number of experiments with various combinations of parameters were performed. The results are compared and the recommended strategy is briefly discussed. The best result was produced by EA with OPX and mutation rate 0.4. Słowa kluczowe: Modular shading system, initial conditions, multiobjective optimization, discrete optimization, backtracking, orderbased representation, evolutionary algorithm Afiliacje autorów:
 15p.  
12.  Zawidzki M.^{♦}, Nishinari K.^{♦}, Shading for Building Facade with TwoColor OneDimension RangeTwo Cellular Automata on A Square Grid, Journal of Cellular Automata, ISSN: 15575969, Vol.8, pp.147163, 2013 Streszczenie: A practical application of cellular automata (CA) in the field of Architecture is presented, where onedimensional CA on a square grid drives a modular shading system of a building facade. Examples of three classes: general, semitotalistic and totalistic of twocolor onedimension rangetwo automata are analyzed for potential practical use. The problem of desired change in the opacity of the shading array as a function of the sequence of initial conditions (SIC) is discussed. The ideal SlCs for selected CAs on 12 x 12 cell arrays found by backtracking algorithm are presented. The influenee of the type of boundary conditions (BC) is discussed and a special type of a fixed BC is introduced. The robustness of the system under two types of permanent failure is analyzed  the single cell failure, and deactivation of a single column of cells. Słowa kluczowe: Adaptive architecture, modular shading system, fixed boundary conditions, Robustness, failure Afiliacje autorów:
 15p.  
13.  Zawidzki M.^{♦}, Nishinari K.^{♦}, Modular TrussZ system for selfsupporting skeletal freeform pedestrian networks, Advances in Engineering Software, ISSN: 09659978, DOI: 10.1016/j.advengsoft.2011.12.012, Vol.47, No.1, pp.147159, 2012 Streszczenie: This paper presents the concept of TrussZ (TZ) – a skeletal system for pedestrian traffic which is composed of only two modules and allows the creation of complex threedimensional selfsupporting networks connecting any number of terminals in a given environment. TZ is intended as a universal, feasible and practical system for newly designed situations and most importantly, for retrofitting, especially where the use of heavy equipment is impossible or uneconomic. Słowa kluczowe: TrussZ, Modular skeletal system, Selfsupporting structure, Organic design, Discrete structural optimization, Retrofit pedestrian link, Pathfinding with backtracking Afiliacje autorów:
 25p.  
14.  Zawidzki M.^{♦}, Tateyama K.^{♦}, Nishikawa I.^{♦}, The constraints satisfaction problem approach in the design of an architectural functional layout, ENGINEERING OPTIMIZATION, ISSN: 0305215X, DOI: 10.1080/0305215X.2010.527005, pp.124, 2011 Streszczenie: A design support system with a new strategy for finding the optimal functional configurations of rooms for architectural layouts is presented. A set of configurations satisfying given constraints is generated and ranked according to multiple objectives. The method can be applied to problems in architectural practice, urban or graphic design—wherever allocation of related geometrical elements of known shape is optimized. Although the methodology is shown using simplified examples—a single story residential building with two apartments each having two rooms—the results resemble realistic functional layouts. One example of a practical size problem of a layout of three apartments with a total of 20 rooms is demonstrated, where the generated solution can be used as a base for a realistic architectural blueprint. The discretization of design space is discussed, followed by application of a backtrack search algorithm used for generating a set of potentially ‘good’ room configurations. Next the solutions are classified by a machine learning method (FFN) as ‘proper’ or ‘improper’ according to the internal communication criteria. Examples of interactive ranking of the ‘proper’ configurations according to multiple criteria and choosing ‘the best’ ones are presented. The proposed framework is general and universal—the criteria, parameters and weights can be individually defined by a user and the search algorithm can be adjusted to a specific problem. Słowa kluczowe: Architecture, design support, architectural layout optimization, multiobjective, discrete Optimization, CSP Afiliacje autorów:
 30p. 
Lista ostatnich monografii
1. 559  Zawidzki M., Discrete Optimization in Architecture  Building Envelope, Springer, pp.1124, 2017 
2. 560  Zawidzki M., Discrete Optimization in Architecture  Architectural & Urban Layout, Springer, pp.1105, 2016 
3. 561  Zawidzki M., Discrete Optimization in Architecture  Extremely Modular Systems, Springer, pp.1121, 2016 
Prace konferencyjne
1.  Zawidzki M., Szklarski J., Singlebranch TrussZ Optimization Based on Image Processing and Evolution Strategy, PARENG2017, International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering, 20170530/0531, Pécs (HU), DOI: 10.4203/ccp.111.28, pp.28, 2017 Streszczenie: TrussZ (TZ) is a skeletal system for creating freeform pedestrian ramps and ramp networks among any number of terminals in space. TZ structures are composed of four variations of a single basic unit subjected to affine transformations (mirror reflection, rotation and combination of both). This paper presents a new approach to the optimization of the layout of a singlebranch TrussZ (STZ) in constrained environment (E). The problem is formulated as follows: create an STZ from a start (sP) to end point (eP) without selfintersections and collisions with two obstacles. This is a multicriterial optimization problem where three independent objectives are subjected to minimization: the total number of modules (n), the “reaching error” to eP and the “overlapping error”. All three criteria are weighted and aggregated to a single cost function (CF). The calculation of CF is based on image processing of rendered geometry of individual STZs in E. The optimization is performed by populationbased classic heuristic method  Evolution Strategy (ES). The computation of CF is the most time consuming, however, its parallelization is rather straightforward. Two parallelization methods are presented: distribution over Wolfram Lightweight Grid and application of general purpose graphical processing units (GPGPUs) with the use of CUDA platform. Słowa kluczowe: Extremely Modular System, TrussZ, discrete optimization, image processing, rasterization, GPU, CUDA, Mathematica, Wolfram Lightweight Grid. Afiliacje autorów:
 
2.  Zawidzki M., Szklarski J., Preliminary Optimization of PipeZ Reconfiguration, PARENG2017, International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering, 20170530/0531, Pécs (HU), DOI: 10.4203/ccp.111.27, pp.27, 2017 Streszczenie: PipeZ (PZ) is a parametric design system which comprised of a congruent modules (PZM) allows the creation of complex threedimensional, singlebranch structures which can be represented by mathematical knots. Once the geometrical parameters are set for the PZM, the shape of PZ is controlled solely by relative twists of the PZMs in a sequence. Therefore each PZM has one degree of freedom (1DOF). This paper presents the preliminary optimization of PZ reconfiguration from a “straight tube” to a halftorus. Here the displacement of PZMs transverse to the “bending direction” is to be minimized. In other words, it resembles “truing” of a wheel. In the considered case, the PZ is comprised of eight hexagonal PZMs. Thus every PZM can have six possible positions relative to the previous module. The initial (PZI) and target (PZT) configurations are given. Since the timesteps and relative twists are discrete, it is a discrete optimization and has combinatorial nature. The number of possible configurations grows astronomically with the assumed number of timesteps from one position to another and the number of PZMs. However, the optimization algorithm can be naturally parallelized. At first the concept of PZ is outlined, followed by the experiment. The results are illustrated and discussed. Słowa kluczowe: Extremely Modular System, PipeZ, ArmZ, discrete optimization, dihedral rotation, “snakebot”, reconfiguration. Afiliacje autorów:
 
3.  Zawidzki M., Jankowski Ł., Szklarski J., Structural optimization of a fiveunit singlebranch TrussZ modular structure, SMART 2017, 8th Conference on Smart Structures and Materials, 20170605/0608, Madrid (ES), pp.525535, 2017 Streszczenie: TrussZ (TZ) is an Extremely Modular System (EMS). Such systems allow for creation of structurally sound freeform structures, are comprised of as few types of modules as possible, and are not constrained by a regular tessellation of space. Their objective is to create spatial structures in given environments connecting given terminals without selfintersections and obstacleintersections. In an EMS, the assembly, reconfiguration and deployment difficulty is moved towards the module, which is relatively complex and whose assembly is not intuitive. As a result, an EMS requires intensive computation for assembling its desired freeform geometrical configuration, while its advantage is the economization of construction and reconfiguration by extreme modularization and mass prefabrication. TZ is a skeletal modular system for creating freeform pedestrian ramps and ramp networks among any number of terminals in space. TZ structures are composed of four variations of a single basic module (TrussZ module, TZM) subjected to affine transformations (mirror reflection and rotation). The previous research on TZ focused on global discrete optimization of the spatial configuration of modules. This contribution reports on the first attempts at structural optimization of the TZM for a singlebranch TZ. Namely, the internal topology of a TZM and sizing of its elements are subject to optimization. An important challenge is due the fact that TZM is to be universal, i.e., it must be designed for the worst case scenario. There are four variations of each module, and due to symmetries there are thus 4^4 = 256 unique 5unit configurations. The structural performance of all of them needs to be evaluated in terms of a typical structural criterion (the maximum von Mises effective stress), and used for structural optimization at the level of a single TZM. Słowa kluczowe: Extremely Modular System, TrussZ, Structural optimization, Effective stress Afiliacje autorów:
 
4.  Zawidzki M., Jankowski Ł., Multicriterial optimization of geometrical and structural properties of the basic module of a singlebranch TrussZ structure, WCSMO12, 12th World Congress of Structural and Multidisciplinary Optimisation, 20170605/0609, Braunschweig (DE), pp.19, 2017 Streszczenie: TrussZ (TZ) is an Extremely Modular System (EMS). Such systems allow for creation of structurally sound freeform structures, are comprised of as few types of modules as possible, and are not constrained by a regular tessellation of space. Their objective is to create spatial structures in given environments connecting given terminals without selfintersections and obstacleintersections. In an EMS, the assembly, reconfiguration and deployment difficulty is moved towards the module, which is relatively complex and whose assembly is not intuitive. As a result, an EMS requires intensive computation for assembling its desired freeform geometrical configuration, while its advantage is the economization of construction and reconfiguration by extreme modularization and mass prefabrication. Słowa kluczowe: multicriterial optimization, TrussZ, effective stress, modular systems Afiliacje autorów:

Patenty
Numer/data zgłoszenia patentowego Ogłoszenie o zgłoszeniu patentowym  Twórca / twórcy Tytuł Kraj i Nazwa uprawnionego z patentu  Numer patentu Ogłoszenie o udzieleniu patentu  

426131 20180629    Zawidzki M. Modularny system do tworzenia pochylni o dowolnym rzucie i o stałym stopniu nachylenia oraz sposób jego montażu PL, Instytut Podstawowych Problemów Techniki PAN     