Sensitivity analysis and shape optimization in nonlinear solid mechanics

Purpose – A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities. Design/methodology/approach – A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation. Findings – Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency. Originality/value – Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.

Geometrical nonlinearity arises from the continuous change of shape under increasing loads on thin or long span members, and the materials nonlinearity due to the nonlinear behavior of the materials composing the structural element. In this study the finite element method is used to analyze fibrous reinforced concrete slabs, material and geometric nonlinearities were considered. Concrete is represented by degenerate quadratic thick shell element. Concrete behavior in compression is modeled by elastic perfectly plastic or strain hardening plasticity. A tensile strength criterion is used to follow crack initiation and propagation. A parabolic stress degradation function, which is based on the fracture energy concept, is used to model the tension stiffening effect of steel fibers. The reinforcement is considered as smeared within the concrete layers, and assumed as either an elastic perfectly plastic material or as an elastic-plastic material with linear strain hardening. Von Karman assumptions which is based on the total Lagrangian approach is used for the geometric nonlinearity. The numerical results using for both normal and lightweight fiber reinforced concrete slabs were predicted satisfactorily. The average predicted failure load of the investigated cases to the experimental failure load was 0.899 with a standard deviation of 0.092.

Nonlinear analysis of 3-D steel frames

Adequate mechanical properties make stainless steel an excellent construction material for structures. The current stainless steel European structural code EN1993‐1‐4 (2015) is largely based on provisions given for carbon steel EN1993‐1‐1 (2005) and does not establish specific design rules for the global analysis of stainless steel structures. Thus, the same classification criterion applies in both codes for sway and non‐sway structures, as well as providing the same predicting expressions for the amplification of bending moments in basic structures, despite the considerable differences between the mechanical behaviour of these two materials. However, recent numerical research on stainless steel frames with H cross‐sections by Walport et al. (2019) showed that the degradation of stiffness due to material nonlinearities considerably affects the response of stainless steel frames, causing greater deformations and increasing second order effects.In order to contribute to the optimal design of stainless steel structures, this paper investigates the influence of the nonlinear material response of stainless steel alloys on frame‐type structures with Rectangular Hollow Sections based on the experimental results conducted by Arrayago et al. (2019), and the interaction of material and geometric nonlinearities.

Dynamic soil–structure interaction (SSI) represents an interdisciplinary subject characterized by complex geometrical and material nonlinearities affecting both the soil and the structural components of the system under study. In the current practice, SSI problems are often solved using the so-called substructure method based on decomposing the superstructure-foundation-soil system into two subsystems whose response is determined independently. The direct method is a more general procedure to tackle SSI problems which is in principle capable of accounting for both soil and structural nonlinearities. Despite this generality, most of the current computational platforms are specialized either for structural applications or for geotechnical applications. Whereas the formers are capable to accurately reproduce the nonlinear response of the structural elements, they are usually poor in modeling soil nonlinear behaviour especially under earthquake loading. On the other hand, the latter do the reverse: they are advanced in modeling soil response but typically rough in reproducing the behaviour of the structural components of the system. Inevitably, all structures interact with the ground and therefore the need to model material and geometric nonlinearities maybe important both in the ground and structural elements. The variation of the stiffness mismatch between soil and structural elements during ground motions is one of the most influential parameters for the assessment of the kinematic component in SSI problems. The variation of the stiffness ratio is mainly governed by material nonlinearity that can develop in either the soil or the structural parts of the system. This paper presents a nonlinear constitutive model for beam elements that is compatible with performance–based earthquake engineering principles because it is capable to simulate the cyclic degradation of mechanical properties and to assess the damage suffered by the structural system. The proposed model is based on distributed plasticity and can be easily implemented in any computational platform allowing the use of an explicit solver. The article describes the theoretical aspects of the model. Validation and application of the model to the solution of a dynamic soil–structure interaction problem where the nonlinear behaviour of geomaterials is coupled with the inelasticity of structural elements, are presented in a companion paper (Andreotti and Lai in Bull Earthq Eng 2017).

Beam-like, architected microstructures (BAMs) can be used in various functional applications, such as micro-electro-mechanical systems (MEMS), energy harvesting, actuation, etc. Designing with high aspect ratios under large deformations, BAMs typically experience severe buckling instability. Therefore, it is of necessity to investigate the mechanical behaviors with respect to the geometric and material nonlinearities. This paper proposes a bilinear size-dependent model using the modified couple stress theory to full formulate the buckling response of the BAMs. Buckling-induced large deformation is studied using the geometric nonlinearity, and the recoverability is investigated based on the material nonlinearity. Experiments and numerical simulations are conducted to validate the theoretical results at the macroscale and microscale, respectively, and satisfactory agreements are obtained. Parametric studies are carried out to study the effects of the material length scale factor-to-thickness ratio χ=lt (i.e., geometric nonlinearity) and plastic factor η (i.e., material nonlinearity). χ and η are further discussed with respect to the effective bending stiffness B¯PAM and volume V¯PAM of the microstructures. The theoretical model presented in this study can be used to predict and tune the mechanical response of BAMs with different architected webbing patterns.

Pulsed power equipment is often characterized by high energy density and field intensity. In the presence of strong electric field intensity, charge accumulation within insulators exacerbates electric field non-uniformity, leading to potential insulation breakdown, thereby posing a significant threat to the safe operation of pulsed power equipment. In this manuscript, we introduce nonlinear composite materials with field-dependent conductivity and permittivity to adaptively regulate the distribution of the pulsed electric field in insulation equipment. Finite-element modeling and analysis of the needle-plate electrodes and high-voltage bushing are carried out to comprehensively investigate the non-uniformity of the distribution of the electric field and the homogenization effect of various nonlinear materials in the presence of pulsed excitations of different timescales. Numerical results indicate that the involvement of nonlinear composite materials significantly improves the electric field distribution under pulse excitations. In addition, variations in the rising time of the pulses affect the maximum electric field intensity within the insulators considerably, but for pulses of nanosecond and microsecond scales, the tendencies are the opposite. Finally, via the simulations of the bushing, we illustrate that some measures proposed for improving the uniformity of the electric field under low frequencies, e.g., increasing the length of the electric field equalization layer and the distance of the underside of the electric field equalization layer from the grounding screen, are still effective for the homogenization of pulsed electric field.

It is critical to handle geometric and material nonlinearities in a stable manner while solving the problems from solid mechanics, such that it results in a converged solution. The present work compares the suitability of generalised displacement control (GDC) and displacement control algorithms (DCA) by solving several 1D and 2D formulations. The ability of these algorithms to handle homogeneous and inhomogeneous deformations is also studied. A novel direct displacement control method (DDCM), coupled with Newton-Raphson method, is proposed and compared with GDC and DCA approaches. Appropriate conclusions are finally drawn based on the successful demonstrations of the numerical results obtained by GDC, DCA and DDCM approaches.

Nonlinear aerostatic stability analysis of suspension bridges

Given the vast advancements in computing power in the last several decades, nonlinear dynamic analysis has gained wide acceptance by practicing engineers as a useful way of assessing and improving the seismic performance of structures. Nonlinear structural analysis software packages give engineers the ability to directly model nonlinear component behavior in detail, resulting in improved understanding of how a building will respond under strong earthquake shaking. One key component, in particular, for understanding the behavior of moment-resisting frames is the plastic rotation of the flexural hinges. Performance-based standards typically use plastic rotation as the primary parameter for defining the acceptance criteria in moment-resisting frames. Since plastic rotation is a key parameter in the seismic damage assessment, the concept as well as the method to calculate this quantity must be understood completely. Though engineers rely on the plastic rotation output from seismic structural analysis software packages to determine acceptable performance, the actual calculation methods used in achieving such plastic rotation quantities usually lay within a so-called “black box”. Based on the outputs obtained from most structural analysis software packages, it can be shown that running an algorithm considering material nonlinearity by itself will produce reasonably accurate results. Moreover, separately running an algorithm considering geometric nonlinearity also can produce accurate results. However, when material nonlinearity is combined with geometric nonlinearity in an analysis, obtaining accurate results or even stable solutions is more difficult. The coupling effect between the two nonlinearities can significantly affect the global response and the local plastic rotation obtained from the analysis and therefore needs to be verified through some analytical means. Yet, the verification process is difficult because a robust analytical framework for calculating plastic rotation is currently unavailable and urgently needed. In view of this gap, an improved analytical approach based on small displacement theory is derived to calculate the plastic rotations of plastic hinges at various locations of moment-resisting frames. Both static and dynamic analysis with nonlinear geometric effects will be incorporated in the derivation. Here the element stiffness matrices are first rigorously derived using a member with plastic hinges in compression, and therefore the coupling of geometric and material nonlinearity effects is included from the beginning of the derivation. Additionally, plastic rotation is handled explicitly by considering this rotation as an additional nonlinear degree-of-freedom. Numerical simulation is performed to calculate the nonlinear static and dynamic responses of simple benchmark models subjected to seismic excitations. Results are compared with various software packages to demonstrate the feasibility of the proposed method in light of the output results among software packages in calculating plastic rotations.

Optimization of complex mechanics simulations with object-oriented software designs

This paper presents a computationally efficient numerical model for the analysis of thin shells based on rotation-free triangular finite elements. The geometry of the structure in the vicinity of the observed triangular element is approximated through a controlled domain consisting of nodes of the observed finite element and nodes of three adjacent finite elements between which a second-order spatial polynomial is defined. The model considers large displacements, large rotations, small strains, and material and geometrical nonlinearity. Material nonlinearity is implemented by considering the von Mises yield criterion and the Levi–Mises flow rule. The model uses an explicit time integration scheme to integrate motion equations but an implicit radial returning algorithm to compute the plastic strain at the end of each time step. The presented numerical model has been embedded in the program Y based on the finite–discrete element method and tested on simple examples. The advantage of the presented numerical model is displayed through a series of analyses where the obtained results are compared with other results presented in the literature.

Generalized quasicontinuum modeling of metallic lattices with geometrical and material nonlinearity and variability

Finite element modelling of cold-formed steel beams under local buckling or combined local/distortional buckling

A finite element framework for the interplay between delamination and buckling of rubber-like bi-material systems and stretchable electronics