Randomized low-rank approximation methods for projection-based model order reduction of large nonlinear dynamical problems

Projection-based nonlinear model order reduction (MOR) methods typically make use of a reduced basis to approximate high-dimensional quantities. However, the most popular methods for computing V , eg, through a singular value decomposition of an m × n snapshot matrix, have asymptotic time complexities of and do not scale well as m and n increase. This is problematic for large dynamical problems with many snapshots, eg, in case of explicit integration. In this work, we propose the use of randomized methods for reduced basis computation and nonlinear MOR, which have an asymptotic complexity of only or . We evaluate the suitability of randomized algorithms for nonlinear MOR and compare them to other strategies that have been proposed to mitigate the demanding computing times incurred by large nonlinear models. We analyze the computational complexities of traditional, iterative, incremental, and randomized algorithms and compare the computing times and accuracies for numerical examples. The results indicate that randomized methods exhibit an extremely high level of accuracy in practice, while generally being faster than any other analyzed approach. We conclude that randomized methods are highly suitable for the reduction of large nonlinear problems.     

Crack evolution law and failure mode of red sandstone under fatigue–creep interaction

To explore the fracture characteristics of rock under the interaction of fatigue load and creep load, fatigue–creep interactive loading experiments were performed on red sandstone with prefabricated cracks. The crack evolution process and failure mode were analyzed using acoustic emission technology and digital image correlation. The results showed that crack growth mainly occurred in the fatigue loading stage; the crack evolution of the sample could be divided into three stages: nucleation and initiation ( ), stable expansion ( ), and unstable fracture ( ). There were distinct differences in the crack propagation modes of the rock samples with different prefabricated crack angles. The relationship between the crack initiation angle and prefabricated crack angle was analyzed based on the maximum circumferential stress theory. Moreover, with an increase in the prefabricated crack angle, the rock sample gradually changed from compression–shear failure to tension–shear failure.     

Fixed-precision randomized low-rank approximation methods for nonlinear model order reduction of large systems

Many model order reduction (MOR) methods employ a reduced basis to approximate the state variables. For nonlinear models, V is often computed using the snapshot method. The associated low-rank approximation of the snapshot matrix can become very costly as m,n grow larger. Widely used conventional singular value decomposition methods have an asymptotic time complexity of , which often makes them impractical for the reduction of large models with many snapshots. Different methods have been suggested to mitigate this problem, including iterative and incremental approaches. More recently, the use of fast and accurate randomized methods was proposed. However, most work so far has focused on fixed-rank approximations, where rank k is assumed to be known a priori. In case of nonlinear MOR, stating a bound on the precision is usually more appropriate. We extend existing research on randomized fixed-precision algorithms and propose a new heuristic for accelerating reduced basis computation by predicting the rank. Theoretical analysis and numerical results show a good performance of the new algorithms, which can be used for computing a reduced basis from large snapshot matrices, up to a given precision ε.     

The analysis for morphological evolution and crystallography of degenerate pearlite in 100Mn13 steel

During approximate 773 K aging treatment of 100Mn13 steel, degenerate pearlite will occur and evolve into lamellar pearlite during growth process. The microstructures of degenerate pearlite and its evolutionary lamellar pearlite are observed by scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The results show that after 748 K, 773 K and 798 K aging, degenerate pearlites occur at grain boundary. At growth front of degenerate pearlite forming at 773 K and 798 K, pearlite presents a morphology of short lamellae of carbide and ferrite, indicating a trend of developing into lamellar pearlite. The higher the temperature is, the more obvious the trend is, and even a conventional lamellar pearlite has developed. However, there is no morphological evolution for degenerate pearlite forming at 748 K aging. Besides, the constituents of degenerate pearlite is identified as M₂₃C₆ and ferrite, and Kurdjumov-Sachs orientation relationship exists between them, (01 )_α//( 1 )_M23C6, [111]_α//[110]_M23C6. This orientation relationship maintains in morphological evolution from degenerate pearlite to lamellar pearlite.     

Toughness enhancement and equivalent initial fracture toughness of cementitious composite reinforced with aligned steel fibres

Based on theoretical analysis and numerical simulation, the impact of steel fibres on the stress intensity factor (SIF) at the crack tip for cementitious composite was studied. The enhanced toughness of steel fibre reinforced cementitious composite (SFRC) in resisting cracks was explained by the decrement of SIF caused by steel fibre inclusions at the crack tip of the composite. The equivalent initial fracture toughness was used to characterize the crack initiation of SFRC. A simplified method for determining the of SFRC was proposed based on a linear regression method. Fracture tests were conducted on three‐point bending notched beams with different steel fibre volume fractions and specimen sizes to study the crack initiation behaviour of aligned steel fibre reinforced cementitious composite (ASFRC). of ASFRC was calculated, and the size effect of was analysed. The results showed that slightly increased with the steel fibre volume fraction and gradually became stable. For the tested specimens, whose heights varied between 40 and 100 mm, the specimen size had little impact on the .     

Optimal design of the side‐sensitive modified group runs (SSMGR) chart when process parameters are estimated

A side‐sensitive modified group runs (SSMGR) chart is commonly examined under the assumption of known process parameters (denoted as Case‐K). However, in practical situations, the process parameters are seldom known, and it is essential to estimate them from in‐control (IC) phase I samples. From the study of the minimum amount of phase I samples, m that is required by the SSMGR chart when the process parameters are estimated (represented as Case‐U), to achieve practically the same performance as that of the Case‐K, it was found that a huge number of IC phase I samples are needed. Due to time and cost factors, accumulating a substantial amount of IC phase I samples is difficult. To circumvent this problem, we study the optimal designs of the Case‐U SSMGR chart, based on both the zero and steady states' average number of observations to signal (ANOS) and expected ANOS (EANOS). Tables containing optimal charting parameters are provided to ease the implementation of the Case‐U SSMGR chart. By using these optimal parameters specifically designed for the Case‐U SSMGR chart, we can obtain similar zero and steady states' ANOS and EANOS performances to that of the Case‐K. The use of the optimal Case‐U SSMGR chart is demonstrated with real data taken from a hard‐bake process in a manufacturing company.     

Microstructure and fatigue performance of friction stir welded joints of 2A12-T4 and 7075-T6 dissimilar aluminum alloy

The objective of this work is to investigate the microstructure and fatigue performance of friction stir welding joint of 2A12-T4 and 7075-T6 dissimilar aluminum alloy. Microstructure shows that the grain growth of heat-affected zone in both sides of joint is noticeable; the grain size of thermo-mechanically affected zone in both sides of joint is deformed and distorted in various degrees; nugget zone is fine equiaxed grain with uniform distribution. Microhardness of joint presents an approximate “W” types distribution on the whole, and the highest hardness value of the joint appears near the center of nugget zone, up to 146.5 HV0.2. Since the joint has the higher the fatigue strength ratio, and the S−N curve of the joint shows a smaller downward trend, the joint has the best fatigue performance and 7075-T6 base metal has the worst fatigue performance. Linear fitting of 7075-T6 base metal is ; linear fitting of 2A12-T4 base metal is ; linear fitting of joint is . The fracture analysis presents that the low-cycle fatigue fracture mechanism of welding joint is quasi-cleavage fracture, and the main mode of high cycle fatigue fracture is the inter crystalline fracture, mixed with ductile fracture.     

Research on corrosion performance of 6061 aluminum alloy in salt spray environment

Salt spray corrosion test was carried out on 6061 aluminum alloy, and quasi-static tensile test at room temperature was carried out on the sample with universal testing machine. The effect of salt spray corrosion on the mechanical properties of 6061 aluminum alloy was studied by scanning electron microscope (SEM), energy dispersive spectrometer (EDS) and electrochemistry. The corrosion rate of 6061 aluminum alloy was quantitatively characterized by different corrosion parameters. It was found that local corrosion of 6061 aluminum alloy occurred in salt spray environment, mainly pitting corrosion and intergranular corrosion. With the increase of corrosion time, the polarization resistance of 6061 aluminum alloy decreases, and the corrosion rate significantly increases. The average corrosion rate and the maximum corrosion rate of 6061 aluminum alloy were characterized by corrosion weight loss and corrosion pit depth. And they can be transformed into each other. The mechanical properties of 6061 aluminum alloy were mainly affected by the depth of corrosion pit. With the increase of corrosion time, the tensile strength and fracture strain decreased, resulting in poor plasticity of the sample. At the same time, the change of elongation of 6061 aluminum alloy can be accurately predicted by the depth of corrosion pit.     

Study of the influence of notch radii and temperature on the probability of failure: A methodology to perform a combined assessment

The fracture assessment of notched components based on cracked components approaches leads to over‐conservative failure predictions. In the research literature, several approaches are proposed to overcome this problem using an apparent fracture toughness, . Nevertheless, most of these approaches are based on deterministic assumptions despite the large and variable scatter exhibited by for different notch radii (ρ) or temperatures (T). This paper proposes a methodology for deriving a probabilistic field including the effect of temperature on the failure of notched components. First, the theory of critical distances is applied to transform each apparent fracture toughness into the equivalent fracture toughness for ρ = 0. Then, the temperature is supposed to act as a scale effect in the Weibull cumulative distribution function of the equivalent fracture toughness, and the corresponding scale effect function is derived. Finally, the applicability of the proposed methodology is illustrated by an example using two ferritic‐pearlitic steels: S275JR and S355J2.     

Influence of artificial aging time on bending characteristics of 6061 aluminum sheet

6061 aluminum alloy has been widely utilized in modern automobile industries with the advantages of low density, good formability and nice machinability. However, the springback phenomena and cross-section deformation problems still remain unsolved. Accordingly, the effect of artificial aging time on the aforementioned bending properties of 6061 aluminum sheet at room temperature were investigated based on the three-point bending test. The mechanical properties of 6061 aluminum alloy was significantly changed by artificial aging treatment, and the bearing capacity of T6-8 h specimen is 2.11 times that of solution heat treatment state specimen when the deflection is the same. With the finite element model verified by experiments, the variation laws of equivalent stress and effective plastic strain of 6061 aluminum sheets in different states were analyzed, and the influence of work hardening exponent n and strength coefficient K on the springback and cross-section deformation were also obtained. The springback increases when the work hardening exponent n decreases gradually, and the uneven deformation increases with the increase of strength coefficient K. With the increase of strength coefficient K, the warpage and the thinning rate increases gradually. As the artificial aging time extends, the springback and cross-section deformation increase.     

Determination of dielectric properties of natural fiber reinforced polymer composite using adaptive neuro fuzzy inference system

In this study, the dielectric properties of polylactic acid reinforced with natural fibers are objectified by using adaptive neuro fuzzy inference system (ANFIS). This method is considered to be economically feasible as compared to fabricating dielectric samples and measuring dielectric properties with apt operating conditions and advanced equipment. Also, past research has focused primarily on polymer dielectric properties which does not involve any natural fiber inclusion, such that this research will focus on producing ANFIS models for natural fibers which will then be used to calculate dielectric permittivity ( ), dielectric loss ( ) with respect to frequency dependencies. Furthermore, experimental results of dielectric constants and losses of polylactic acid (PLA) based composites will be analyzed from past research which used physical techniques to fabricate composites and will be compared with the results from ANFIS models. It was found that error computation of both the properties are found to be low and the percentage difference in output data is considered less. In addition, the ANFIS neural network had predicted most of the data that was trained, and the crisp output values were found to correlate with the experimental dielectric properties which makes this prediction model possible to use as an alternative approach to fabricating composites and testing.     

Crack initiation and propagation characteristics of a dual-phase Mg–Li alloy under high-cycle and very-high-cycle fatigue regimes

Ultralight Mg–Li alloys are promising aerospace materials as they are the lightest structural alloys at present; however, their fatigue behaviors remain to be explored. This work focuses on the fatigue strength and crack initiation behavior of an extruded dual-phase Mg–Li alloy (LZ91) under high-cycle and very-high-cycle fatigue regimes. The fatigue limit of LZ91 alloy at 10⁹ cycles was determined to be 78 MPa, and the fatigue ratio is 0.46. Microstructural characterization demonstrates that fatigue cracks tend to initiate at β-Li phase-enriched regions. The α-Mg phase presents a < > fiber texture with a basal plane that has low deformation in the extrusion direction and acts as an enhanced phase in relation to the β-Li phase. Deformation discrepancies cause localized cyclic plasticity at the Li phase that leads to fatigue crack initiation.     

Location charts based on ranked set sampling for normal and non‐normal processes

Control charts, based on ranked set sampling schemes, had been proposed recently for efficient monitoring of process location. All the proposals in the literature are based on the ideal assumption of normally distributed quality characteristics. No study as of yet investigated the performance of location charts based on ranked set sampling for non‐normal processes. In this study, we investigated the location chart based on simple random sampling (SRS) and three well‐known rank‐based schemes ie, ranked set sampling (RSS), median ranked set sampling (MRSS), and extreme ranked set sampling (ERSS), considering normal and a variety of non‐normal parent distributions. Both heavy‐tailed symmetric and skewed cases have been considered in this study. The performance of the charts is evaluated using average run length (ARL), extra quadratic loss (EQL), and relative ARL (RARL) measures. A real life example is also presented that details the monitoring of pH levels in water for an experiment conducted to study the reproduction of Mysids. The study will help quality practitioners to choose the chart based on an efficient sampling scheme for normal and non‐normal processes.     

Fatigue assessment of as‐built and heat‐treated Inconel 718 specimens produced by additive manufacturing including notch effects

The fatigue behaviour of notched and unnotched specimens produced by additively manufactured Inconel 718 were analysed in the as‐built and heat‐treated conditions. The surfaces display high roughness and defects acting as fatigue initiation sites. In the as‐built condition, fine subgrains were found, while in in the heat‐treated state, the subgrains were removed and the dislocation density recovered. SN‐curves are predicted based on tensile properties, hardness and defects obtained by fractography, using the ‐method.     

Energy conservation during remeshing in the analysis of dynamic fracture

The analysis of (dynamic) fracture often requires multiple changes to the discretisation during crack propagation. The state vector from the previous time step must then be transferred to provide the initial values of the next time step. A novel methodology based on a least-squares fit is proposed for this mapping. The energy balance is taken as a constraint in the mapping, which results in a complete energy preservation. Apart from capturing the physics better, this also has advantages for numerical stability. To further improve the accuracy, Powell-Sabin B-splines, which are based on triangles, have been used for the discretisation. Since continuity of the displacement field holds at crack tips for Powell-Sabin B-splines, the stresses at and around crack tips are captured much more accurately than when using elements with a standard Lagrangian interpolation, or with NURBS and T-splines. The versatility and accuracy of the approach to simulate dynamic crack propagation are assessed in two case studies, featuring mode-I and mixed-mode crack propagation.     

Optimized identification of earlywood and latewood stiffnesses in loblolly pine in simulated experiments

Knowledge of local mechanical behaviour of wood is especially important as silvicultural practices are modified to allow wood to compete as a relevant material in high technology applications. Challenges associated with identification of local mechanical behaviour have resulted in simplified test geometries designed to determine one or two constitutive parameters. The objective of this work was to design and simulate an entire experiment developed to simultaneously identify the earlywood and latewood orthotropic stiffnesses in loblolly pine in a single specimen and load geometry. The virtual experiment was capable of evaluating optimal orthotropy orientation for reduced identification errors and indicating most favourable choices for data smoothing filters and identification methodology. Additionally, certain ring spacing and latewood percentages were shown to produce large errors, but those combinations are unlikely to occur naturally. The simulation was able to identify $Q_{11},Q_{22}$, and $Q_{66}$ with approximately $\pm 10\%$ error; the $Q_{12}$ error was larger with more scatter. The methodology presented here contributes to the best practices available for heterogeneous stiffness identification.     

Microstructure evolution and texture formation of 16 % chromium ferritic stainless steel following simulated batch annealing treatments in mass production

Batch annealing technique is mainly used in industry for improving productivity as a few steel coils were stacked and heated in a bell-type furnace. The microstructure evolution, texture formation and mechanical properties of 16 % chromium ferritic stainless steel under different simulated batch annealing and subsequent cold-rolled annealing conditions were investigated in this work. Results showed that batch annealing process applied in mass production could not produce fully recrystallized and homogenously equiaxed grains even at very high temperatures up to 900 °C for 30 hours. With increased batch annealing temperature, a large number of chromium carbides precipitated in ferrite, while some unstable Fe-carbide precipitates were gradually dissolved. Relatively lower cold-rolled annealing temperature (830 °C) led to finer grains and superior mechanical properties of 16 % chromium ferritic stainless steel. Increased batch annealing temperature improved the intensity of {111}//normal direction γ-fiber textures at the expense of other orientations including {hkl}<110> α-fiber, {334}<4 3>, thus improving the formability of ferritic stainless steel.     

Effect of electromagnetic stirring and solution treatment on microstructure and properties of AZ31 magnesium alloy

Effect of electromagnetic stirring and solution treatment on the microstructural evolution and mechanical properties of AZ31 alloy are investigated in this study. Results show that the mechanical properties of AZ31 alloy are substantially increased after electromagnetic stirring and solution treatment. The tensile strength and elongation are 178 MPa and 19.6 % for AZ31 alloy prepared by electromagnetic stirring after solution treatment. A large number of twins and low angle boundaries are observed in the specimen after heat treatment, since the compressive stress on ( ) crystal plane become larger during quenching. The dislocation arrays, which produced by slip or climbing of dislocations, join together to form low angle boundaries.     

An algorithm for triangulating smooth three-dimensional domains immersed in universal meshes

We describe an algorithm to recover a boundary-fitting triangulation for a bounded C²-regular domain immersed in a nonconforming background mesh of tetrahedra. The algorithm consists in identifying a polyhedral domain ω_h bounded by facets in the background mesh and morphing ω_h into a boundary-fitting polyhedral approximation Ω_h of Ω. We discuss assumptions on the regularity of the domain, on element sizes and on specific angles in the background mesh that appear to render the algorithm robust. With the distinctive feature of involving just small perturbations of a few elements of the background mesh that are in the vicinity of the immersed boundary, the algorithm is designed to benefit numerical schemes for simulating free and moving boundary problems. In such problems, it is now possible to immerse an evolving geometry in the same background mesh, called a universal mesh, and recover conforming discretizations for it. In particular, the algorithm entirely avoids remeshing-type operations and its complexity scales approximately linearly with the number of elements in the vicinity of the immersed boundary. We include detailed examples examining its performance.