This state-of-the-art paper reports the last ten year results, obtained by an informal research group completed of participants of some Polish universities at the Institute of Computer Methods in Civil Engineering (now Institute of Computational Civil Engineering) of the Cracow University of Technology, and supervised by the author of the paper. After a short introduction and brief discussion of ANNs basic ideas, the activities in five areas are described: i) ANNs as a new independent computational tool for the analysis of C&SE problems, ii) neural networks in FEM/ANN hybrid systems developed for the C&SE problems analysis, iii) various problems analyzed by ANNs, iv) modifications of BPNNs (Back-Propagation Neural Networks) and new learning methods, as well as other ANNs than those applied in problems mentioned above, v) promotion of ANNs. The representative six selected study cases are discussed: 1) concrete fatigue failure, 2) buckling of cylindrical shells with geometrical imperfections, 3) acceleration response spectra, 4) reliability of a plane frame, 5) hybrid updating of a thin-walled beam FE model, 6) hybrid identification of equivalent material in a perforated strip. Some general conclusions on prospects of ANNs applications in C&SE are given at the end of the paper.
Keywords: artificial neural networks (ANNs), back-propagation neural network (BPNN), finite element method (FEM), hybrid FEM/ANN system, civil and structural engineering (C&SE), Cracow University of Technology.

The Bodner-Partom elastic-visco-plastic constitutive equations [1] were used for numerical analysis of inelastic problems. This rate-dependent model makes it possible to describe elastic, plastic and viscous processes in metals, including temperature and continuum damage effects. The adaptive finite element method [2] was applied to approximate solution of the governing equations with two a posteriori error estimates that control accuracy of time and space discretization of displacements and internal variables. The paper addresses a further development of the methodology proposed by the author in previous works [3,4] and used in [5]. We present here certain additional theoretical background and propose a novel strategy of adaptation as well as verify the method of solution transfer. [1] S.R. Bodner, Y. Partom, J. Appl. Mech., 42: 385-389, 1975. [2] L. Demkowicz, J.T. Oden, W. Rachowicz, O. Hardy, Comp. Meth. Appl. Mech. Engng., 77: 79-112, 1989. [3] W. Cecot, W. Rachowicz, CAMES, 7: 479-492, 2000. [4] W. Cecot, Int. J. Num. Meth. Engng., 61: 2139-2158, 2004. [5] W. Cecot, Analysis of selected in-elastic problems by h-adaptive finite element method, Cracow University of Technology, 2005.
Keywords: h-adaptive finite element method, error estimate, elastic-visco-plasticity.

Wood exhibits an intrinsic structural hierarchy. It is composed of wood cells, which are hollow tubes oriented in the stem direction. The cell wall is built up by stiff cellulose fibrils which are embedded in a soft polymer matrix. This structural hierarchy is considered in a four-step homogenization scheme, predicting the macroscopic elastic behavior of different wood species from tissue-specific chemical composition and microporosity, based on the elastic properties of nanoscaled universal building blocks. Special attention is paid to the fact that the fibrils are helically wound in the cell wall, at an angle of 0°-30°, generally denoted as microfibril angle. Consideration of this microfibril angle in the continuum micromechanics model for wood is mandatory for appropriate prediction of macroscopic stiffness properties, in particular of the longitudinal elastic modulus and the longitudinal shear modulus. The presented developments can be readily extended to the prediction of poroelastic properties, such as Biot and Skempton coefficients.
Keywords: wood, continuum micromechanics, anisotropic elasticity, wood cell wall, experimental validation.

This paper deals with the second-order computational homogenisation of a heterogeneous material undergoing small displacements. Typically, in this approach a representative volume element (RVE) of nonlinear heterogeneous material is defined. An a priori given discretised microstructure is considered, without focusing on detailed specific discretisation techniques. The key contribution of this paper is the formulation of equations coupling micro- and macro-variables and the definition of generalized boundary conditions for the microstructure. The coupling between macroscopic and microscopic levels is based on Hill's averaging theorem. We focus on deformation-driven microstructures where overall macroscopic deformation is controlled. In the end a numerical example of a thin layer shear is presented.

The concept of experimental data interpretation in mechanics using Dirac function is presented. The objective is to find general differential equation, which may be used to approximate the stress field sought. Application of this method may convert problem from general spline to variational one. Basic idea has been presented earlier in [1] where both theoretical data and results of experiments have been taken into account. Here, the method has been used to solve a 2D problem. Numerical tests performed for both generated and experimental data proved its usefulness. [1] W. Karmowski, J. Orkisz, Inverse Problems in Engineering Mechanics, pp 61-70, Springer-Verlag, 1993.

The object under study reported on in this paper was an aortic valve based on a natural aortic valve as the original. Simulations were carried out to examine the functioning of a valve which was loaded with varying pressure until a buckling of the leaflets and a full opening of the valve were observed. The aim of the study was the optimal choice of the geometric and mechanical parameters for the class of construction assumed for analysis.

A mathematical model describing coupled heat, moisture and salt transport in porous materials is presented. Salt dissolved in water can be transported due to various mechanisms: dispersion caused by the salt concentration gradient, and advection resulting from the capillary pressure gradient. The influence of salt on the physical properties of water such as density and dynamic viscosity is also considered. The isotherms of water sorption are modified to take into account both osmosis and effects of the salt presence on the surface tension and contact angle. Salt precipitation in the state of thermodynamic equilibrium between dissolved and crystallized salt is also considered. Finally, the model equations were discretized in space by means of FEM and the HMTRA-SALT software was developed. An example concerning a wall drying process was numerically solved to show the robustness of the code.
Keywords: porous materials, salt transport, hydrodynamic dispersion, coupled transport.

Kalman filtering is used as a learning method for the training of Feed-forward Layered Neural Networks (FLNN) and Recurrent LNNs (RLNN). These networks were applied to the simulation of hysteresis loops obtained by the experiment on a cable-in-conduit superconductors by the test carried out in a cryogenic press [1]. The training and testing patterns were taken from nine selected, characteristic hysteresis loops. The formulated FLNN: 4-4-5-1 gives the computer simulation of higher accuracy than the standard network FLNN: 4-7-5-1 discussed in [2]. [1] A. Nijuhuis, N.H.W. Noordman, H.H.J. ten Kate, Mechanical and Electrical Testing of an ITER CS1 Model Coil Conductor under Transverse Loading in a Cryogenic Press, Univ. of Twente, 1998. [2] M. Lefik, in Fusion Engineering and Design, 105-117, Elsevier, 2002.
Keywords: Kalman filtering, neural network, simulation, conductor, hysteresis loops.

Simulation of the long term behavior of a metacarpal bone with a prosthesis is presented with the help of an evolutive 3D finite element model taking into account "the stress shielding" phenomena. The same model allows to improve the shape of the prosthesis.
Keywords: prosthesis, bone remodeling, optimisation, iterative method.

In the paper presented is an application of the physically based global method (PBGM) to a posteriori estimation of experimental data error. It is proposed here to build data error measures by spanning a high quality physically reasonable smoothing fit to data and treat it as a reference field for error estimation in a very similar way it is done in the postprocessing type error estimates used widely in FE or meshless methods, where the higher order (superconvergent) solutions are used for building error estimates (post-processing type of error estimators). The new technique is different from classical methods of experimental data error estimation as it provides non-statistical estimates of the data error and as such it may be applied to a wider range of problems, including cases when only a single data set is available (e.g., destructive testing). And because the new approach builds the estimates while performing its standard physically based global-type approximation, it fully integrates other features of the PBGM approach like data interpolation, extrapolation or differentiation. In the paper the whole PBGM approach is presented, including the concept of the method formulated for the case of analysis of residual stress in railroad rails, discretisation with MFDM, then several PBGM a posteriori error estimates are introduced and results for test problems (benchmark and actual data) are shown.
Keywords: hybrid methods, physically based approximation, meshless finite differences method, smoothing of experimental data, error estimation techniques.

Frost heave in soils is a common phenomenon in cold regions, yet the previous efforts toward its mathematical description did not result into a generally accepted model. The model described in this paper is based on the concept of porosity rate function, which characterizes well the heaving phenomenon in variety of soils. The concept is simple enough so that it can be easily incorporated in numerical methods. The description of the model is followed by brief considerations of energy transfer and phase change. Calibration results are shown, and the model is implemented to solve a practical boundary value problem. The influence of thermal insulation on the performance of a retaining wall with frost-susceptible backfill is discussed.
Keywords: soil freezing, frost heave, constitutive model, heat transfer, phase change, retaining wall.

Possible yielding of the cross-section of a structure, which may arise as a result of external actions or the (micro)defects, might significantly decrease the safety margin of the considered structure [1]. Since the cross-section yielding affects the structure stiffness, the dynamic characteristics (eigenvalues and eigenvectors) might be significantly different then the ones of the original structure. The measurement of the changes of the dynamic parameters may provide the information necessary to identify the load causing the yielding of the cross-section and further the yielding index (which may be calculated when the load causing the yielding is know) enables the evaluation of the structure safety margin. This paper presents the application of Artificial Neural Networks (ANN) [2,3] in the identification of the load casing partial yielding of simply-supported beam and one- or two-column frames. [1] W.F. Chen, H. Zhang, Structural Plasticity: Theory, Problems and CAE Software, Springer-Verlag, 1991. [2] S. Haykin, Neural Networks. A Comprehensive Foundation, Prentice-Hall, 1999. [3] Z. Waszczyszyn, L. Ziemiański, CAMES, 13: 125-159, 2006.
Keywords: finite element method, identification, dynamics, artificial neural networks.

In the paper the structure optimization system based on the surface remodeling is presented. The base of algorithm formulation was the trabecular bone surface remodeling phenomenon leading to optimization of the trabecular net in the bone as well as the design with optimal stiffness principle. The closed system including Finite Element mesh generation, decision criterion for structure adaptation and Finite Element Analysis in parallel environment are presented. The issues concerning the use of the tool for the mechanical design are discussed. Some results of computations, using special prepared software are presented.
Keywords: biomechanics, structural optimization.

The paper is concerned with a class of generalized structural optimization problems, in which geometrical nonlinearities play an important role in a response of dynamically loaded structure. Forced, steady-state periodic vibrations of linear elastic frame and beam structures are considered. Both, viscous and complex modulus damping models are used. Using the adjoint variable method, sensitivity operators with respect to variation of stiffness, damping and mass parameters, as well as loading and support conditions are derived. The loading corresponds to an excitation induced by a rotational machine founded on vibro-isolation. The forms of response functionals expressed in displacements are discussed. Numerical examples of frame structures illustrate the theory and demonstrate the accuracy of the derived sensitivity operators.
Keywords: sensitivity analysis, optimal design, second order geometric effects, structural dynamics, vibrations.

The focus of this paper is the application of two nonlinear regression models in the context of Bayesian inference to the problem of failure prediction of concrete specimen under repeated loads based on experimental data. These two models are compared with an empirical formulae. Results on testing data show that both models give better point predictions than empirical formulae. Moreover, Bayesian regression approach makes it possible to calculate prediction intervals (error bars) describing the reliability of the models predictions.
Keywords: Bayesian inference, concrete, fatigue failure, Gaussian process regression, feed-forward neural network.

The influence of fluid phase on soil instabilities is investigated using the modified Cam-clay model within a two-phase description. Spurious mesh dependence of finite element results is prevented by a gradient enhancement of the model. The results of numerical tests for one-phase and two-phase model are compared. The influence of permeability on the stabilizing role of the fluid phase is assessed.
Keywords: two-phase medium, finite element method, plasticity, Cam-clay model, gradient regularization, localization.

Large deflection analysis of laminated composite plates is considered. The Galerkin method along with Newton-Raphson method is applied to large deflection analysis of laminated composite plates with various edge conditions. First order shear deformation theory and von Kármán type nonlinearity are utilized and the governing differential equations are solved by choosing suitable polynomials as trial functions to approximate the plate displacement functions. The solutions are compared to that of Chebyshev polynomials and finite elements. A very close agreement has been observed with these approximating methods. In the solution process, analytical computation has been done wherever it is possible, and analytical-numerical type approach has been made for all problems.
Keywords: Galerkin method, large deflection, composite plates.