**A Machine learning-based constitutive model for nonlinear analysis via Finite Element Method**

Álefe Freitas Figueiredo

CILAMCE 2020 - XLI Iberian Latin American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

This paper addresses a machine learning technique in the context of constitutive modelling. Since it has been proven that multilayer perceptrons with the backprogapation algorithm are capable of approximating any class of functions, studies have been developed with the objective of using it as approximation functions for the nonlinear behaviour of complex material media. This is only possible because neural networks have a powerful adaptability, capability of learning and generalizability. In this context, a multilayer perceptron is trained with stress-strain results from a nonlinear analysis via finite element method with Mazars material in order to develop a

neural network-based constitutive model. This implementation is carried out with the help of a recognized machine learning package in order to obtain more accurate results. To validate the proposed constitutive model, the results obtained through the multilayer perceptron are compared with the ones of the finite element numerical analysis.

**A meshfree cell-based smoothed radial point interpolation method for damage problems**

Samir Silva Saliba

CILAMCE 2020 - XLI Iberian Latin American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

Meshfree methods belonging to the class of Smoothed Point Interpolation Methods (S-PIM) have been shown to provide certain advantages with respect to the standard finite element method (FEM), when dealing with physically nonlinear problems. The present work extends the Cell-Based Smoothed Radial Point Interpolation Method with polynomial reproduction (CS-RPIMp), originally proposed for linear problems, to the case of damage models. The weakened-weak (W2) formulation and peculiar integration scheme which this method is based on have been extended to nonlinear damage models. Some numerical examples of nonlinear problems with different kinds of boundary conditions, performed using different strategies for support nodes selection based on the T-schemes (T4-, T6/3- and T2L-schemes), are presented, aiming to point out the accuracy, convergence and efficiency of the Cell-Based Smoothed Radial Point Interpolation Method with polynomial reproduction in comparison with the standard finite element method.

**Cohesive crack propagation simulated by different SGFEM strategies**

Thaianne S. de Oliveira, Felício B. Barros, Bruna C. Campos

CILAMCE 2020 - XLI Ibero-Latin-American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

The present work aims to evaluate the performance of the Stable Generalized Finite Element Method (SGFEM), a relatively new approach that derives from a simple modification of enrichment functions used in Generalized/eXtended Finite Element Method (G/XFEM), in the analysis of a three-point bending test. For this, different crack propagation simulations are performed using the standard Heaviside Function, its linear modification as proposed by Gupta et al. and a version that employs a stabilization parameter, presented in Wu and Li. A cohesive crack model is considered and linear elastic material is assumed for the numerical experiments. Equilibrium paths, as well as the scaled condition numbers (SCNs), calculated at each step, are evaluated by SGFEM and compared with the results obtained by G/XFEM. This work is related to a proposal of expansion of the INSANE (INteractive Structural ANalysis Environment) system, an open source project developed at the Structural Engineering Department of the Federal University of Minas Gerais. This platform has enabled the resources that allowed the analysis and discussions carried out in this work.

**Comparison of Generalized/eXtended Finite Element Methods for quasi-brittle media cracking problems**

Larissa Novelli

CILAMCE 2020 - XLI Iberian Latin American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

This work presents a comparative study of the application of the Generalized/eXtended Finite Element Method (GFEM) in the solution of cracking problems. Different strategies are performed: Polynomial enrichment strategy with the GFEM and numerical enrichment strategy with and without Stable Generalized Finite Element Method (SGFEM) procedure. The numerical enrichment strategy is based on global-local analysis. For this strategy, the nonlinear analysis is performed in the global problem and a local problem is solved in the end of each converged step. The local solution is used as numerical enrichment for next incremental step of the global problem. This local problem, solved with a fine mesh, is a subdomain of the global problem in the cracking region of the problem. For the application of the polynomial enrichment strategy, the same subdomain of global problem is enriched with prescribed polynomial functions. The smeared cracking model is used as elastic-degradation constitutive model to simulate the behavior of quasi-brittle media. The implementations have been performed in the INSANE (Interactive Structural Analysis Environment) system, a free software developed at Department of Structural Engineering of Federal University of Minas Gerais. Numerical example of a two-dimensional problem (2D) is presented for validation and comparison of the strategies. Besides, the results are compared with experimental data and reference solutions obtained via classical Finite Element Method (FEM).

**Comparison of smoothed point interpolation methods in linear static problems**

Samir Silva Saliba

CILAMCE 2020 - XLI Iberian Latin American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

Smoothed point interpolation methods (S-PIM), when compared with the standard finite element method (FEM) in the study of static linear problems, present interesting results due certain characteristics of their formulation. In general, meshfree methods have been shown to be more accurate and efficient. The proposed study aims to present a comparison between the Node-Based Smoothed Radial Point Interpolation Method with polynomial reproduction (NS-RPIMp), Edge-Based Smoothed Radial Point Interpolation Method with polynomial reproduction (ES-RPIMp) and Cell-Based Smoothed

Radial Point Interpolation Method with polynomial reproduction (CS-RPIMp), applied to static linear-elastic problems. Several numerical simulations, performed using differents strategies for support nodes selection based on the T-schemes (T3-, T4-, T6/3- and T2L-schemes), are presented, providing a relevant amount of results that allow to understand the behaviour of each one of these methods and schemes with respect to the accuracy and convergence rate.

**Damage models for micromorphic continuum**

Pâmela Daniela Nogueira

CILAMCE 2020 - XLI Iberian Latin American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

Numerous materials, although they appear macroscopically homogeneous, usually present a heterogeneous microstructure that directly influences the structural behavior. The phenomenological approach followed by the classical continuum mechanics does not individually accounts for this nfluence, which can be significant, for example, in cases where the structure or the specimen under analysis is small compared to its microstructure or when the material media has a complex microstructure. Within the framework of continuum mechanics, so-called generalized continuum theories are particularly suited to deal with the above issues incorporating the microstructural behavior on the formulation. The micromorphic theory is included in this general class of generalized continua and, more specifically, in the group that incorporates additional degrees of freedom to the material particles. Another aspect of generalized continua is its ability to address the issue of localization in quasi-brittle materials modeled as elastic-degrading media as a result of its non-local character. In order to allow the representation of quasi-brittle media with the micromorphic continuum theory, this work presents a formulation for scalar-isotropic damage models for a micromorphic continuum in the constitutive models framework of the INSANE system, initially conceived for classical media and later expanded for the micropolar continuum. This implementation is based on the tensorial format of a unified constitutive models formulation and homogenization techniques to obtain the micromorphic constitutive relations.

**Nonlinear analysis of three-point bending beams via global-local Generalized Finite Element Method**

Anelize Borges Monteiro, Felício Bruzzi Barros, Roque Luiz da Silva Pitangueira, Samuel Silva Penna

CILAMCE 2020 - XLI Ibero-Latin-American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

The Generalized Finite Element Method (GFEM) was developed in order to overcome some limitations inherent to the Finite Element Method (FEM), related to the defects propagation, presence of large deformations or even in the description of high gradients of state variables. The GFEM enriches the space of the polynomial FEM solution with a priori known information based on the concept of Partition of Unit (PoU). Certain obstacles of nonlinear analysis can be mitigated with the GFEM, and damage and plasticity fronts can be accurately represented. In this context, the global-local approach to the GFEM (GFEM global-local) was proposed. The success of its application to problems of Linear Elastic Fracture Mechanics is already proven, but its extension to the simulation of collapse of structures made of quasi-brittle media is still a vast field to be researched. Here, a coupling strategy is presented based on the global-local GFEM to describe the deterioration process of quasi-brittle media, such as concrete, in the context of Continuous Damage Mechanics. The numerical solution used to enrich the global problem, represented by a coarse mesh, is obtained through physically nonlinear analysis performed only in the local region where damage propagation occurs, represented by constitutive models. The linear analysis of the global region is performed considering the incorporation of local damage through the global-local enrichment functions and damage state mapped from local problem. Numerical examples of three-point bending notched concrete beams have been presented to evaluate the performance of the proposed approach and the obtained results were compared with the experimental results and with the ones obtained with standard GFEM. Two constitutive models were applied to represent the concrete in the local region: smeared crack model and microplane model.

**Nonlinear Cyclic Analysis with Unloading-Reloading Laws based on Moving Focal Point**

Lívia Ramos Santos Pereira, Samuel Silva Penna

CILAMCE 2020 - XLI Ibero-Latin-American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

Constitutive models are mechanical-mathematical formulations that describe the behavior of a material. For a complete representation of the medium, these models must be able to reproduce the processes of loading, unloading and reloading. Usually, models for quasi-brittle materials present deterioration of the elastic modulus, while residual strains are not verified. Models for ductile materials, although, have their elastic modulus preserved during unloading and that results in permanent strains. In real materials analysis what is observed is a composition of these two behaviors. The literature presents analytical models to incorporate both hypotheses, developed from the concept of focal point. Traditional formulations usually work with a fixed focal point. Thus, in order to expand existing formulations, this work presents an elastic degradation model associated with cyclic stress-strain laws according to a moving focal point, which varies depending on the deterioration of the material medium.

**On the numerical integration in G/XFEM analysis for physically nonlinear problems and cohesive crack**

Bruna C. Campos, Felício B. Barros, Samuel S. Penna

CILAMCE 2020 - XLI Ibero-Latin-American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

This work presents a novel methodology to deal with some drawbacks related to crack propagation in physically nonlinear problems in the context of the Generalized Finite Element Method (GFEM). The GFEM associates Finite Element Method shape functions to local approximation functions that describe, for example, the discontinuity from Fracture Mechanics cohesive crack problems. In this case, numerical integration must be adapted for properly dealing with the non-polynomial integrand of the weak form of the boundary value problem. A common alternative to consider the discontinuity is the employment of the subdivision of elements, in the integration points are changed according to the cited strategy. Although a very efficient procedure in linear problems, it leads to the loss of the state constitutive variables history, responsible for indicating the degradation level in physically nonlinear materials. A new strategy is here proposed, based on the nonlocal approach to recover the evolution of the state constitutive variables in the integration points. A numerical example is provided to validate and prove the efficiency of the proposed methodology. The computational implementation and analysis are performed in the open source software INSANE (Interactive Structural Analysis Environment), developed by the Structural Engineering Department of Federal University of Minas Gerais.

**Phase-field modelling of diffuse fracture with FEM**

Hugo Mouro Leão

CILAMCE 2020 - XLI Iberian Latin American Congress on Computational Methods in Engineering - 2020

## Resumo (em inglês)

In damage models, cracks are considered in a smeared way, without any geometric representation of the region where the crack takes place, and the energy released is used in crack growth that is controlled by the energy fracture parameter. Phase-field models consider a diffuse and smooth crack that belongs to a certain volume region, where a function describes the crack density. In that region each point has a field variable that quantifies the material degradation. The phase-field techniques allows to detect crack paths and its bifurcation without having a pre-existent crack. The purpose of this work is to present some phase-field models implemented in the INSANE (Interactive Structural ANalysis Environment system) software, an structural analysis open-source software developed by at the Structural Engineering department (DEES) of the Federal University of Minas Gerais (UFMG). This work opens a new research line inside the INSANE Project. Some preliminary results will be presented.