Macroscopic Finite Element Model Research Articles

Multiscale modelling of microstructure evolution, and local solidification behaviours of the AlSi10Mg build component in laser powder bed fusion process

Laser powder bed fusion (L-PBF) process is an additive manufacturing process, in which rapid heating and cooling takes place. The microstructure and the solidification behaviour of L-PBF components have significant impacts on their mechanical properties. The solidification takes place in a very short duration which is difficult to capture experimentally, therefore the finite-element (FE) simulation is the best feasible solution. In this paper, to study the evolution of the AlSi10Mg alloy’s microstructure during the L-PBF process, a model composed of macroscopic finite element (FE) model and microscopic phase field (PF) model was built. This model is suitable to predict a more realistic thermal behaviour involving the possible physical phenomena and the thermal behaviour for various heat inputs was calculated. The PF model was then updated with these results in order to simulate the chemical compositions’ concentration field and solidified microstructure. According to the simulation’s results, the primary dendrite arm spacing (PDAS) is significantly influenced by the laser line energy. To validate the FE model along with the Phase field (PF) model, the results were compared with the analytical and experimental findings; and the FE model and PF results exhibited good agreement with the experimental results. Additionally, the solute distribution analysis demonstrated that during the solidification process, micro-segregation developed and the process parameters affected the micro-segregation. Finally, the specimens were subjected to metallographic and element distribution analyses using an energy-dispersive spectrometer (EDS) to compare the results with those from the PF simulation and establish the relationship between the process parameters.

Elucidating the influence mechanisms of splat cooling on microstructure evolution in friction stir welding of 2195 Al–Li alloy by multi-scale simulation

Al–Li alloy has been widely used in the manufacture of aerospace aircraft structural parts due to its high specific strength and excellent comprehensive performance. Friction stir welding (FSW), as a solid-state joining technique, generates comparatively lower heat generation in contrast to fusion welding. However, even with this reduced heat input, the welded joints of 2195 Al–Li alloy still exhibit noticeable thermal softening. FSW is very sensitive to welding heat input, and the thermal cycle of high peak temperature coarsens the grain in the weld zone, thereby reducing joint properties. To overcome these problems, external cooling can be applied to reduce peak temperature and increase cooling speed, improving the performance of FSW joints. In order to study the mechanism of external cooling on the microstructure evolution of joints, this work established a multi-scale simulation of splat cooling assisted friction stir welding (SCaFSW) of 2195 Al–Li alloy. The temperature change and material deformation data in SCaFSW were calculated in the macroscopic finite element model and used for further Monte Carlo microstructure evolution simulation, and compared with the conventional FSW process. Results reveal that the splat cooling significantly reduces the area of the high temperature region of the workpiece, decreasing the temperature at the same location 10 mm behind the tool by approximately 50 °C compared to FSW. While the local temperature field around the tool pin is not affected by the splat cooling, which mainly accelerates the cooling rate of the welded joint and reduces the plastic strain. In addition, the dislocation density in the weld nugget zone (WNZ) of SCaFSW joint is higher than that in conventional FSW, which promotes the process of grain nucleation and finally refines the grain in the WNZ. The simulated average grain size and thermal cycling results are in good agreement with the experimental results.

Multi-scale pore model construction and damage behavior analysis of SiCf/SiC composite tubes

SiCf/SiC composite materials are essential for spacecraft thermal protection, determining the safety of spacecraft. However, their complex structures and intricate fabrication processes have led to an unclear understanding of their failure mechanisms, limiting their application. In this study, circumferential compression experiments were conducted, and X-ray computed tomography (X-CT) was used to analyze the distribution characteristics of pore spaces. The experimental results indicate that damage manifests as secondary fracture phenomena based on temporal evolution characteristics. Specifically, after experiencing performance degradation of 24.73–60.96%, the material still maintained basic stability, exhibiting a performance recovery of 31.55–36.1% under the applied load until failure. To predict the dynamic damage behavior of materials at multiple scales, a multi-scale method combining statistical and microscopic approaches was proposed. A micro representative volume element (RVE) random fiber pore distribution model was established based on random processes and random medium theory. Finally, considering the CT data and the principle of minimum potential energy, a macroscopic finite element model of the micro-pores' structural characteristics in a three-dimensional wound tube was reconstructed. The results indicate that cracks initiate around pore defects and gradually propagate to form crack bands. The model closely matches the macroscopic damage and microscopic characteristics of the material. This work provides a new approach for the numerical simulation of pore defects in such materials.

Finite element modeling of the magnetic property behaviors of electrical steels in rotating fields considering their crystallographic textures

In magnetic device modeling, the crystallographic textures and microstructures of electrical steels critically influence their magnetic properties. Yet, these aspects are often neglected or inadequately represented in finite element (FE) models, leading to inaccuracies in predicting electrical steel performance, especially in complex electromagnetic environments. This paper presents a multi-scale modeling approach, integrating a macroscopic FE model of a magnetic device with a microstructural scalar permeability model that accounts for the crystallographic textures of electrical steels. This approach demonstrates proficiency in predicting nonlinear anisotropic permeability and capturing magnetic anisotropy of electrical steels within a rotational tester, especially when anisotropy is predominantly governed by crystallographic texture and the angles between the B and H vectors are minimal. Furthermore, the model effectively captures the magnetic anisotropy in complex magnetic fields and geometric configurations of a permanent magnet motor. A comparative analysis of this model is conducted against three other material models: an isotropic model, a two-axis model, and a general-vector model. These models are based on different approaches, ranging from a single non-linear BH curve to comprehensive vector BH curves covering all directions. Among them, the general-vector model, reliant on extensive 2D vector BH measurements, is the most accurate but requires substantial experimental data.

Torsional actuation and vari-stiffness characteristics of SMA/basalt hybrid braided composite tubes

Shape memory alloy hybrid composite (SMAHC) tubes have the potential to actively control the torsional stiffness of structures. In this work, shape memory alloy (SMA) wires orthogonally braided with basalt fiber were heated by electricity to generate the torsional torque of the SMAHC tube, and then the torsional stiffness of the tube was measured. The macroscopic finite element model of the SMAHC tube was established to assist in the analysis of the torsional stiffness variation before and after heating. According to the experimental results, the actuating torsional ability was affected by the phase transformation recovery force of SMA. The active control of torsional stiffness depends on the change of SMA modulus, the magnitude and direction of SMA wires’ recovery force, and the thermodynamic properties of matrix at different temperatures. Then an evaluation model for the torsional stiffness control effect was proposed. These research results can be used for the adaptive control of the stiffness and vibration characteristics of rotating composite structures, and provide a novel and effective method for the control of structural torsional characteristics with lightweight effects.

A Multi-Scale Investigation to Predict the Dynamic Instabilities Induced by Frictional Contact

We propose a new variational formulation to model and predict friction-induced vibrations. The multi-scale computational framework exploits the results of (i) the roughness measurements and (ii) the micro-scale contact simulations, using the boundary element method, to enrich the contact zone of the macroscopic finite element model of rubbing systems with nominally flat contact boundaries. The resulting finite elements at the contact interface of the macroscopic model include (i) a modified normal gap and (ii) a micro-scale description of the contact law (i.e., pressure gap) derived by solving the frictionless contact problem on a rough surface indenting a rigid half-plane. The method is applied to a disc brake system to show its robustness in comparison with classical deterministic formulations. With respect to the traditional complex eigenvalues analysis, the proposed multi-scale approach shows that the inclusion of roughness significantly improves the results at low frequencies. In this panorama, any improvement of dynamic instabilities predictions should be based on an uncertainty analysis incorporating roughness combined with other parameters such as friction coefficient and shear moduli of the pads, rather than on roughness itself.

Deep learning framework for multiscale finite element analysis based on data-driven mechanics and data augmentation

In this study, a deep learning framework for multiscale finite element analysis (FE2) is proposed. To overcome the inefficiency of the concurrent classical FE2 method induced by the repetitive analysis at each macroscopic integration points, the distance-minimizing data-driven computational mechanics is adopted for the FE2 analysis. Macroscopic strain and stress data are directly assigned to the material points of the macroscopic finite element models without constitutive model. Here, the macroscopic problems are solved with the offline macroscopic material genome database, without solving the microscopic problems simultaneously. The novelty of the proposed approach lies in using a deep neural network to enable adaptive sampling points without prior knowledge of the specific mechanical problem. The proposed data augmentation framework updates the sampling points gradually using the distance minimization algorithm with the mechanistic constraints, including the equilibrium and compatibility equations. Particularly, the deep neural network plays a crucial role as a guide in the phase space sampling process, facilitating the efficient use of sparse data. Thus, this method shows high feasibility and significantly improves the computational efficiency of offline computing.

Hybrid discrete-continuum multiscale model of tissue growth and remodeling.

Tissue growth and remodeling (G&R) is often central to disease etiology and progression, so understanding G&R is essential for understanding disease and developing effective therapies. While the state-of-the-art in this regard is animal and cellular models, recent advances in computational tools offer another avenue to investigate G&R. A major challenge for computational models is bridging from the cellular scale (at which changes are actually occurring) to the macroscopic, geometric-scale (at which physiological consequences arise). Thus, many computational models simplify one scale or another in the name of computational tractability. In this work, we develop a discrete-continuum modeling scheme for analyzing G&R, in which we apply changes directly to the discrete cell and extracellular matrix (ECM) architecture and pass those changes up to a finite-element macroscale geometry. We demonstrate the use of the model in three case-study scenarios: the media of a thick-walled artery, and the media and adventitia of a thick-walled artery, and chronic dissection of an arterial wall. We analyze each case in terms of the new and insightful data that can be gathered from this technique, and we compare our results from this model to several others. STATEMENT OF SIGNIFICANCE: This work is significant in that it provides a framework for combining discrete, microstructural- and cellular-scale models to the growth and remodeling of large tissue structures (such as the aorta). It is a significant advance in that it couples the microscopic remodeling with an existing macroscopic finite element model, making it relatively easy to use for a wide range of conceptual models. It has the potential to improve understanding of many growth and remodeling processes, such as organ formation during development and aneurysm formation, growth, and rupture.

Microstructure and Solute Concentration Analysis of Epitaxial Growth during Wire and Arc Additive Manufacturing of Aluminum Alloy

Microstructure and solute distribution have a significant impact on the mechanical properties of wire and arc additive manufacturing (WAAM) deposits. In this study, a multiscale model, consisting of a macroscopic finite element (FE) model and a microscopic phase field (PF) model, was used to predict the 2319 Al alloy microstructure evolution with epitaxial growth. Temperature fields, and the corresponding temperature gradient under the selected process parameters, were calculated by the FE model. Based on the results of macroscopic thermal simulation on the WAAM process, a PF model with a misorientation angle was employed to simulate the microstructure and competitive behaviors under the effect of epitaxial growth of grains. The dendrites with high misorientation angles experienced competitive growth and tended to be eliminated in the solidification process. The inclined dendrites are commonly hindered by other grains in front of the dendrite tip. Moreover, the solute enrichment near the solid/liquid interface reduced the driving force of solidification. The inclined angle of dendrites increased with the misorientation angle, and the solute distributions near the interface had similar patterns, but various concentrations, with different misorientation angles. Finally, metallographic experiments were conducted on the WAAM specimen to validate the morphology and size of the dendrites, and electron backscattered diffraction was used to indicate the preferred orientation of grains near the fusion line, proving the existence of epitaxial growth.

Macroscopic and mesoscopic biomechanical analysis of the bone unit in idiopathic scoliosis

To investigate the effects of postoperative fusion implantation on the mesoscopic biomechanical properties of vertebrae and bone tissue osteogenesis in idiopathic scoliosis, a macroscopic finite element model of the postoperative fusion device was developed, and a mesoscopic model of the bone unit was developed using the Saint Venant sub-model approach. To simulate human physiological conditions, the differences in biomechanical properties between macroscopic cortical bone and mesoscopic bone units under the same boundary conditions were studied, and the effects of fusion implantation on bone tissue growth at the mesoscopic scale were analyzed. The results showed that the stresses in the mesoscopic structure of the lumbar spine increased compared to the macroscopic structure, and the mesoscopic stress in this case is 2.606 to 5.958 times of the macroscopic stress; the stresses in the upper bone unit of the fusion device were greater than those in the lower part; the average stresses in the upper vertebral body end surfaces were ranked in the order of right, left, posterior and anterior; the stresses in the lower vertebral body were ranked in the order of left, posterior, right and anterior; and rotation was the condition with the greatest stress value in the bone unit. It is hypothesized that bone tissue osteogenesis is better on the upper face of the fusion than on the lower face, and that bone tissue growth rate on the upper face is in the order of right, left, posterior, and anterior; while on the lower face, it is in the order of left, posterior, right, and anterior; and that patients' constant rotational movements after surgery is conducive to bone growth. The results of the study may provide a theoretical basis for the design of surgical protocols and optimization of fusion devices for idiopathic scoliosis.

An FE-DMN method for the multiscale analysis of thermomechanical composites

We extend the FE-DMN method to fully coupled thermomechanical two-scale simulations of composite materials. In particular, every Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN). Such a DMN serves as a high-fidelity surrogate model for full-field solutions on the microscopic scale of inelastic, non-isothermal constituents. Building on the homogenization framework of Chatzigeorgiou et al. (Int J Plast 81:18–39, 2016), we extend the framework of DMNs to thermomechanical composites by incorporating the two-way thermomechanical coupling, i.e., the coupling from the macroscopic onto the microscopic scale and vice versa, into the framework. We provide details on the efficient implementation of our approach as a user-material subroutine (UMAT). We validate our approach on the microscopic scale and show that DMNs predict the effective stress, the effective dissipation and the change of the macroscopic absolute temperature with high accuracy. After validation, we demonstrate the capabilities of our approach on a concurrent thermomechanical two-scale simulation on the macroscopic component scale.

Macroscopic simulation model for laser cutting of carbon fibre reinforced plastics

Laser cutting of carbon fibre reinforced plastics (CFRP) has shown promising potential as an alternative to conventional manufacturing processes. Laser cutting has major benefits of contactless and therefore wear-free machining and high automation potential. The main challenge is to reduce the heat input into the material during the process. Excessive temperatures cause damage within the surrounding matrix material and could locally modify the structural properties of the CFRP. For industrial use it is necessary to be able to predict the resulting temperature fields. To gain knowledge of the temperature distribution during the process, a three-dimensional macroscopic finite element model is developed using ANSYS simulation software. Transient-thermal analyses are performed and the material removal process is implemented via the element-death technique. Simulations are run for a unidirectional composite structure and different cutting speeds. The resulting temperatures are compared to experimental data.

Modeling the fire response of reactive powder concrete beams with due consideration to explosive spalling

Reactive powder concrete (RPC) is susceptible to fire-induced spalling because of its dense microstructure and lower permeability compared with normal-strength concrete and high-strength concrete, which may compromise the structural integrity and load-carrying capacity of RPC members significantly. A macroscopic finite-element model was therefore extended to track the fire behavior of RPC beams, especially fire-induced spalling. New sorption isotherms were used to identify the mass of liquid water in RPC at elevated temperatures. Furthermore, an improved spalling criterion was proposed, based on the biaxial strength theory of RPC, to judge if the spalling occurs in RPC members. Additionally, the effect of the improved and previous spalling criteria (based on the uniaxial strength theory) on the fire performance prediction of RPC beams was studied. It was found that the improved spalling criterion can provide more-accurate spalling and fire performance prediction for RPC beams since it considers the effect of both the stresses induced by pore pressure, thermal gradients and applied load and the change in concrete strength caused by the spatial stress superposition.

An FE–DMN method for the multiscale analysis of short fiber reinforced plastic components

In this work, we propose a fully coupled multiscale strategy for components made from short fiber reinforced composites, where each Gauss point of the macroscopic finite element model is equipped with a deep material network (DMN) which covers the different fiber orientation states varying within the component. These DMNs need to be identified by linear elastic precomputations on representative volume elements, and serve as high-fidelity surrogates for full-field simulations on microstructures with inelastic constituents.We discuss how to extend direct DMNs to account for varying fiber orientation, and propose a simplified sampling strategy which significantly speeds up the training process. To enable concurrent multiscale simulations, evaluating the DMNs efficiently is crucial. We discuss dedicated techniques for exploiting sparsity and high-performance linear algebra modules, and demonstrate the power of the proposed approach on an injection molded quadcopter frame as a benchmark component. Indeed, the DMN is capable of accelerating two-scale simulations significantly, providing possible speed-ups of several magnitudes.

Modelling and experimental observation of the deposition geometry and microstructure evolution of aluminum alloy fabricated by wire-arc additive manufacturing

In this paper, a model composed of a macroscopic finite element (FE) model coupled with a microscopic phase field (PF) model was constructed to investigate the Al alloy microstructure evolution during wire arc additive manufacturing (WAAM). First, the relationships between single layer process parameters and the resulting deposition geometries (width and height) were investigated. A three-dimensional FE model was built to calculate the thermal distribution and temperature gradient under different process parameters. These results were then fed into a PF model to obtain the microstructure corresponding to each set of process parameters. A bridge was constructed by this method, starting from the WAAM process parameters and yielding the microstructure throughout solidification. The simulated results showed that the primary dendrite arm spacing (PDAS) decreased slightly with current increases, but the substrate moving speed had a more obvious impact on PDAS, which rose significantly with increased substrate speed. Finally, metallographic examination and energy dispersive spectroscopy tests were conducted on specimens fabricated by WAAM with different process parameters, to verify the simulated PF results. The simulated morphology and size of dendrites were in good agreement with experimental results.

Simulating the effect of fabric bending stiffness on the wrinkling behaviour of biaxial fabrics during preforming

A macroscopic finite element model has been established to investigate the forming-induced wrinkling behaviour for bi-axial fabrics. Results indicate that using a linear bending model with a constant bending stiffness produces unrealistic wrinkle patterns in the fabric plies. A non-linear bending model produces more accurate forming induced wrinkle patterns compared to experimental data, since the bending stiffness parameter is varied as a function of the applied forming load to account for the onset of fibre buckling. Areas of high in-plane shear are more likely to induce out-of-plane wrinkles, indicating a positive correlation between wrinkling onset and shear deformation. A new methodology has been developed to quantitatively evaluate the severity of fabric wrinkles based on the FE simulation results. The distance between the surface of the preform and the mould tool is used to locate areas with out-of-plane defects, using the principal curvature to isolate wrinkles from areas of fabric bridging (poor conformity).

Multiscale modelling of microstructure, micro-segregation, and local mechanical properties of Al-Cu alloys in wire and arc additive manufacturing

Abstract Wire and Arc Additive Manufacturing (WAAM) is a combination of welding and AM technologies and uses an electric arc as the heat source to melt welding wire. The mechanical properties of WAAM components are greatly dependent on the solidification behavior and microstructure. In this paper, a model composed of a macroscopic finite element (FE) model combined with a microscopic phase field (PF) model was constructed to investigate the Al alloy microstructure evolution during WAAM. First, the FE model was implemented to calculate the temperature gradient and solidification speed under different processing parameters. These results were then fed into the PF model to simulate the microstructure and concentration field of chemical compositions. The simulated results indicate that the moving speed of the substrate has a pronounced impact on the primary dendrite arm spacing (PDAS), which grew significantly as the moving speed increased, while the PDAS decreased slightly upon increasing the electric current. Furthermore, the solute distribution analysis showed that micro-segregation formed during the solidification process, and that current and substrate speed had evident impact on the micro-segregation ratio. Finally, metallographic experiments, energy-dispersive spectrometer analysis for element distribution, and miniature tensile tests were conducted on the specimens to validate the PF simulated results and obtain the relationship between process parameters and mechanical properties.

Experimental and multi-scale numerical investigation of ultra-high performance fiber reinforced concrete (UHPFRC) with different coarse aggregate content and fiber volume fraction

This study investigates the influence of the coarse aggregate content and steel fiber volume fraction on the mechanical properties of UHPFRC. On the meso-scale, according to Mori-Tanaka homogenization method and progressive damage theory, the equivalent damage constitutive model of UHPFRC composite is established. On the macro-scale, the mesoscopic equivalent properties of UHPFRC material are input into the finite element model corresponding to the test for numerical simulation, and the mesoscopic material properties are coupled with the macroscopic finite element model to study the influence of mesoscopic components of UHPFRC on its macroscopic mechanical properties. The compression, tension and four-point bending numerical simulation results of UHPFRC show that the predicted results of the multi-scale numerical model are in good agreement with the test results. It is also found that with the increase of fiber volume fraction, the compressive strength, tensile strength and four-point bending strength of UHPFRC increase first and then decrease. The improvement effect of coarse aggregate on the properties of UHPFRC is not as obvious as that of fiber, and the increase of fiber volume fraction has a significant enhancement effect on the ductility and post peak toughness of UHPFRC slab.

Revisiting the demagnetization curves of Dy-diffused Nd-Fe-B sintered magnets

The grain boundary diffusion process (GBDP) is now widely used to increase coercivity in Nd-Fe-B sintered magnets with a more efficient use of heavy rare earth elements (Dy, Tb). This process leads to a typical core-shell structure for the grains consisting of (Nd,Dy)2Fe14B shells at the outer grain regions and Nd2Fe14B cores. The thickness of the (Nd,Dy)2Fe14B shells decreases from the diffusion surface to the magnet core. This inhomogeneous distribution in Dy content gives rise to a coercivity gradient within the magnet and leads therefore to a reduced squareness of the demagnetization curve. The purpose of this work is to provide a quantitative understanding of the influence of composition profiles after GBDP on the shape of the demagnetization curve of Nd-Fe-B sintered magnets diffused with the Dy63Co37 (at. %) intermetallic compound. SEM/X-EDS analyses along the Fisher diffusion model allow the estimation of the Dy concentration in grains and at different depths. Then, after ascribing to the grains some critical values for the switching field that are related to the local Dy content, a macroscopic finite element model is implemented to provide a better understanding of the grain reversal sequence in the graded magnets tested in closed-circuit. Grain reversal patterns show that demagnetization starts from the less coercive grains in the magnet core, remains constricted in this zone thanks to a shielding effect from the external surface, and then propagates towards outer layers via magnetostatic interactions. When the coercivity gradient is large, the coercivity of the whole magnet measured in closed-circuit could be 100–200 kA/m lower than the value expected without considering magnetostatic interactions, suggesting that the shielding effect from the diffusion affected layers could be limited and counterbalanced by magnetostatic interactions.

Pseudo-dynamic response and analytical evaluation of blind bolted CFT frames with BRBs

This paper presents the experimental and analytical studies on the blind bolted end plate concrete-filled steel tube (CFT) composite frames with buckling-restrained braces (BRBs). A series of pseudo-dynamic tests (PDTs) on two 2/3-scaled two-story, one-bay blind bolted end plate CFT (BECFT) composite frames with BRBs showed that the novel dual system exhibited good hysteretic behavior and high ductility. The moment capacity, initial rotational stiffness, rotation capacity and the mathematical moment-rotation model of a BECFT composite joint were systematically analyzed and modified to some extent. Meanwhile, the shear force-deformation relation of a CFT panel zone was suggested and checked for its applicability. Furthermore, the macroscopic finite element (FE) model incorporating semi-rigid characteristic of the joint and shear response of the panel zone was developed by OpenSees program. The numerical analytical results would match better with experimental data when the FE model took the gusset plate stiffness into account in comparison with other models neglecting this case. Both experimental and FE analytical results indicated that the BRBs were in elastic state to provide great lateral stiffness and resistance under small earthquakes, and they yield first to stably dissipate energy under severe earthquakes. The combination of BECFT composite frames with BRBs will extend the feasibility of the novel dual system in high- or even super high-rise buildings.