Partial Differential Equations (PDEs) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the ...behavior of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best-known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate on applications that have an interest for Computational Mechanics. Most contributions explore this possibility have adopted a collocation strategy. In this work, we concentrate on mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. In order to prove the concepts, we deal with several problems and explore the capabilities of the method for applications in engineering.
•Proof of concept for the possibility of approximating the solution of BVPs using concepts and tools coming from deep machine learning.•The energy is the basis for the construction of the loss function.•The approximation space is defined by the architecture of the neural network.•The approach is applied to several engineering problems including linear elasticity, elastodynamics, nonlinear hyperelasticity, plate bending, piezoelectricity and phase field modeling of fracture.
•Sensitivity analysis (SA) approach is used to quantify the effects of correlated input parameters on model outputs.•Penalized spline regression model is used to approximate complex data.
We provide ...a sensitivity analysis toolbox consisting of a set of Matlab functions that offer utilities for quantifying the influence of uncertain input parameters on uncertain model outputs. It allows the determination of the key input parameters of an output of interest. The results are based on a probability density function (PDF) provided for the input parameters. The toolbox for uncertainty and sensitivity analysis methods consists of three ingredients: (1) sampling method, (2) surrogate models, (3) sensitivity analysis (SA) method. Numerical studies based on analytical functions associated with noise and industrial data are performed to prove the usefulness and effectiveness of this study.
Quantitative measurements of both the copy number and spatial distribution of large fractions of the transcriptome in single cells could revolutionize our understanding of a variety of cellular and ...tissue behaviors in both healthy and diseased states. Single-molecule fluorescence in situ hybridization (smFISH)-an approach where individual RNAs are labeled with fluorescent probes and imaged in their native cellular and tissue context-provides both the copy number and spatial context of RNAs but has been limited in the number of RNA species that can be measured simultaneously. Here, we describe multiplexed error-robust fluorescence in situ hybridization (MERFISH), a massively parallelized form of smFISH that can image and identify hundreds to thousands of different RNA species simultaneously with high accuracy in individual cells in their native spatial context. We provide detailed protocols on all aspects of MERFISH, including probe design, data collection, and data analysis to allow interested laboratories to perform MERFISH measurements themselves.
We present an isogeometric thin shell formulation for multi-patches based on rational splines over hierarchical T-meshes (RHT-splines). Nitsche’s method is employed to efficiently couple the patches. ...The RHT-splines have the advantages of allowing a computationally feasible local refinement, are free from linear dependence, possess high-order continuity and satisfy the partition of unity and non-negativity. In addition, the C1 continuity of the RHT-splines avoids the rotational degrees of freedom. The good performance of the present method is demonstrated by a number of numerical examples.
•We present a multi-patch isogeometric large deformation thin shell formulation based on RTH splines. It is an extension of our previous work on RHT-spline shells to large deformations and multiple patches. The coupling is based on Nitsche’s method and allows also coupling of a shell to a solid model.•Furthermore, we present a stress recovery technique to drive the adaptive h-refinement procedure in isogeometric thin structures.•The method is validated for several linear and non-linear benchmark problems including the pinched cylinder and hemispherical shell, a wind turbine rotor accounting for large deformations, a hemispherical shell with a stiffener and a pinched cylinder considering large deformations.
An extended isogeometric element formulation (XIGA) for analysis of through-the-thickness cracks in thin shell structures is developed. The discretization is based on Non-Uniform Rational B-Splines ...(NURBS). The proposed XIGA formulation can reproduce the singular field near the crack tip and the discontinuities across the crack. It is based on the Kirchhoff–Love theory where C1-continuity of the displacement field is required. This condition is satisfied by the NURBS basis functions. Hence, the formulation eliminates the need of rotational degrees of freedom or the discretization of the director field facilitating the enrichment strategy. The performance and validity of the formulation is tested by several benchmark examples.
•An extended isogeometric element formulation (XIGA) for analysis of through-the-thickness cracks in thin shell structures.•XIGA can reproduce the singular field near the crack tip and the discontinuities across the crack.•NURBS basis possesses C1-continuity required by Kirchhoff–Love theory without additional rotational degrees of freedom.
This article presents original work combining a NURBS-based inverse analysis with both kinematic and constitutive nonlinearities to recover the applied loads and deformations of thin shell ...structures. The inverse formulation is tackled by gradient-based optimization algorithms based on computed and measured displacements at a number of discrete locations. The proposed method allows accurately recovering the target shape of shell structures such that instabilities due to snapping and buckling are captured. The results obtained show good performance and applicability of the proposed algorithms to computer-aided manufacturing of shell structures.
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This paper presents an overview of the theories and computer implementation aspects of phase field models (PFM) of fracture. The advantage of PFM over discontinuous approaches to fracture is that PFM ...can elegantly simulate complicated fracture processes including fracture initiation, propagation, coalescence, and branching by using only a scalar field, the phase field. In addition, fracture is a natural outcome of the simulation and obtained through the solution of an additional differential equation related to the phase field. No extra fracture criteria are needed and an explicit representation of a crack surface as well as complex track crack procedures are avoided in PFM for fracture, which in turn dramatically facilitates the implementation. The PFM is thermodynamically consistent and can be easily extended to multi-physics problem by ‘changing’ the energy functional accordingly. Besides an overview of different PFMs, we also present comparative numerical benchmark examples to show the capability of PFMs.
•Development in phase field models and the computer implementation is reviewed.•The theories on phase field modeling are systematically summarized.•Representative numerical examples are presented for different fracture problems.
•A phase field model for fracture in heterogeneous structure.•A hybrid hierarchical/concurrent multiscale method for fracture in polymer-matrix composites.•A phase field model for matrix and ...interphase fracture in polymer-matrix composites.•Extraction of fracture related material properties for various input parameters, particularly for the interphase zone.
We predict the macroscopic tensile strength and fracture toughness of fully exfoliated nano silicate clay epoxy composites accounting for the interphase behavior between the polymeric matrix and clay reinforcement. A phase field approach is employed to model fracture in the matrix and the interphase zone of the polymeric nanocomposites (PNCs) while the stiff clay platelets are considered as linear elastic material. The effect of the interphase zones, e.g. thickness and mechanical properties (Young’s modulus and strain energy release rate) on the tensile strength, and fracture parameters of the composite is studied in detail. The dissipation energy due to fracture in the PNCs is extracted for different thicknesses and properties of the interphase zones. We show through numerical experiments that the interphase thickness has the most influence on the tensile strength while the critical strain energy release rate of the interphase zones affects the dissipation energy depending on the interphase zone thickness.
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