# Special Finite Elements with Adaptive Strain Field on the Example of One-Dimensional Elements
## 1 Introduction  
The first step in the finite element method (FEM) is the discretization of the domain into finite elements. The meaning of this process is essential especially if the influence of the results is concerned. To ensure that the simulation of the original domain is suitable to perform the calculations, such parameters as shape, size, type, and configuration of the elements have to be taken into consideration. Another parameter is the number of the finite elements, which has significant influence of the computational effort. In practice of structural modeling using FEM [1; 2; 3] very often it is necessary to define domains with areas of significantly different geometrical and material characteristics. The obvious solution is to divide these areas into subregions (sub-areas) and describe them with separate finite elements. This approach is natural for FEM, however it can be computationally expensive (a large number of finite elements, complicated discretization algorithms, etc.). In particular it refers especially to materials with high heterogeneity. A typical example of the occurrence of sub-areas with significant stiffness differences is the analysis of physically non-linear processes. In such a case, stiffness in the analysed points can even reach the extremal values, e.g., yielding or cracking. Simultaneously the changes in stiffness are not followed by the adjustment of the strain field. This is one of the main methodological reasons of non-linear analysis error generation. In this case, it could be reasonable to use special type of elements called \"multi-area elements\", where sub-areas with different stiffness parameters can be combined and completed with a self-adaptive procedure of deformation field. It means that one special multi-area element can be used to describe larger (wider) parts of structure even with different geometry or material parameters. Special multi-area elements have been developed to deal with problems in analysis of structures with different geometrical and stiffness parameters for example inclusions or accidental damages in form of various holes, etc. Moreover multi-area elements can be used in analysis of constructions with openings, for example wall panels with openings for doors or windows. The main goal of using such an element is the limitation of element and nodes number in comparison with the fine mesh finite element model [4; 5]. Currently, in the literature the significant number of scientific papers is dedicated to searching for a new finite elements that ensure the appropriate accuracy of the results and simultaneously reduce the number of unknowns in the computational model. They are often called dedicated elements, developed in order to analyse a specific physical phenomenon. Such type of elements are exactly the special multi-area elements, formulated in order to analyse structures and structural systems containing areas of different geometrical and stiffness characteristics. Generally, searching for dedicated elements used in the numerical analysis of building structures is carried out at different levels of observation scales. These issues concern the macroscale and microscale. For example, macromodeling technique is used for large structures and consists in describing by finite element a fragment of a construction with a complex structure, for example part of a brick wall containing several bricks and a mortar between them, like in [6; 7]. Similar approach was represented by [PERSON] and [PERSON] [8]. Their plate rectangular element has been used in analysis of shear walls with openings. The stiffness matrix of such an element has been developed by subtracting the stiffness of opening from the stiffness of whole plane of stress element without opening. The main disadvantage of this method was the increase of calculation error due to increase of the openings size. Another type of element used in the analysis of structures with different stiffness areas are super elements presented by [PERSON] and [PERSON] [9].