Low-density Elements Research Articles

Structural topology optimization of three-dimensional multi-material composite structures with finite deformation

In this work, an explicit three-dimensional (3D) topology optimization approach is presented for multi-material composite structures accounting the finite deformation effect. The proposed method employs different sets of 3D Moving Morphable Voids (MMVs) to identify each phase material, resulting in explicit geometric descriptions of the optimized composite structures and a reduction in the number of design variables. The decoupling between the topology description and finite element analysis of composite structures enables the removal of redundant degrees of freedom, thereby mitigating the convergence issue of finite deformation analysis caused by low-density elements and leading to significant computational savings. Numerical experiments of composite structures with two- and three-phase materials are optimized to validate the accuracy and efficiency of the proposed method. The results demonstrate that, under finite deformation, the distribution of each phase material in optimized 3D composite structures is significantly affected by the amplitude of external loads, and the optimized layout could be quite different from its counterpart under small deformation assumption.

Finite strain topology optimization with nonlinear stability constraints

This paper proposes a computational framework for the design optimization of stable structures under large deformations by incorporating nonlinear buckling constraints. A novel strategy for suppressing spurious buckling modes related to low-density elements is proposed. The strategy depends on constructing a pseudo-mass matrix that assigns small pseudo masses for DOFs surrounded by only low-density elements and degenerates to an identity matrix for the solid region. A novel optimization procedure is developed that can handle both simple and multiple eigenvalues wherein consistent sensitivities of simple eigenvalues and directional derivatives of multiple eigenvalues are derived and utilized in a gradient-based optimization algorithm — the method of moving asymptotes. An adaptive linear energy interpolation method is also incorporated in nonlinear analyses to handle the low-density elements distortion under large deformations. The numerical results demonstrate that, for systems with either low or high symmetries, the nonlinear stability constraints can ensure structural stability at the target load under large deformations. Post-analysis on the B-spline fitted designs shows that the safety margin, i.e., the gap between the target load and the 1st critical load, of the optimized structures can be well controlled by selecting different stability constraint values. Interesting structural behaviors such as mode switching, and multiple bifurcations are also demonstrated.

Lightweight, ultrastrong and high thermal-stable eutectic high-entropy alloys for elevated-temperature applications

Eutectic high-entropy alloys (EHEAs) that combine the advantages of HEAs and eutectic alloys are promising candidates for high-temperature applications. However, currently developed EHEAs still exhibit high densities and low high-temperature strengths, which limit their usage. Here we propose a strategy to design lightweight, strong, and high thermal-stable EHEAs by introducing an extremely stable Heusler-type ordered phase (L21 phase) containing a high-content of low-density elements and constructing a low lattice misfit eutectic-phase interface, which can lead to generate ultrafine and stable lamellar structures and high-density of coherent nanoprecipitates. As a manifestation of this strategy, a novel bulk Al17Ni34Ti17V32 EHEA was designed to consist of L21 and body-centered-cubic (BCC) phases (interlamellar spacing ∼ 320 nm) with a lattice misfit only 2.4%. This alloy has one of the lowest densities (∼ 6.2 g/cm3) among all EHEAs reported previously and exhibits much higher high-temperature hardness and specific yield strengths than most reported refractory HEAs (RHEAs), lightweight HEAs (LWHEAs), EHEAs, and conventional superalloys. This work paves the way to develop light EHEAs with excellent high-temperature properties.

Calculations to introduce some light high entropy alloys based on phase formation rules

Despite high entropy alloys (HEAs), phase formation rules as a simple method have been seldom used to predict the solid solution phases in light high entropy alloys (LHEAs). Some terms and thermodynamic-based parameters are calculated via programing and all 4833 possible quintuple alloys out of proper elements are extracted and categorized in seven alloy groups and fifteen alloy systems with a density less than or equal to 5 g cm −3 defined for LHEAs in this research. As expected, most alloys had low-density elements like Li, Mg, Al, and Ti in their chemical composition. Al 14 Li 11 Mg 35 Ti 15 Zr 25 alloy was identified as the lightest alloy (3.36 g cm −3 ). Various alloys can be obtained and designed with optimizing values for the atomic size difference, electronegativity difference, and other respective terms. All alloy groups and systems are discussed and limitations for alloying methods are described. Al 19 Li 7 Mg 6 Ti 35 Nb 33 LHEA was chosen as a candidate alloy to fabricate by melting and characterization of the as-cast alloy showed the formation of a high hardness single-phase BCC solid solution structure. Acceptable uniformity in the chemical composition of the alloy with a density of 5.31 g cm −3 confirms the relative reliability of phase formation rules. • 4833 novel quintuple LHEAs in 7 alloy groups and 15 alloy systems were introduced. • Al 14 Li 11 Mg 35 Ti 15 Zr 25 was known as the lightest alloy with a ρ th of 3.36 g cm −3 . • Al 19 Li 7 Mg 6 Ti 35 Nb 33 -LHEA was synthesized through the melting and casting method. • Uniform distribution of elements in BCC single solid solution phase was identified. • High hardness and density of 5.31 g cm −3 were measured in as-cast LHEA.

Microstructure development and mechanical performance of Al2CrFeMnTi light-weight high entropy alloy

Light-weight high entropy alloys (LWHEAs) are considered equiatomic or near-equiatomic alloys consisting of at least five elements. Low-density elements, like Al and Ti, are the key constituents in designing these alloys so that features such as reduced overall density and improved mechanical properties are achieved.A new LWHEAs (Al2CrFeMnTi) is designed, where mechanical alloying followed by subsequent casting is carried out for a successful synthesis. As a result, chemically homogenous samples are fabricated that possess a multiphase microstructure of BCC solid solution, C14 Laves phase, and L21 precipitates in the as-cast state. Although the presence of intermetallic phases results in high hardness characterized by nanoindentation tests, the produced alloy demonstrates some level of ductility before failure. This behavior could be linked to the minimal strain hardening of the BCC phase. Additionally, the formation of ultra-fine L21 precipitates within the BCC phase is contributed to the high strength and the modified strain observed in the alloy.

Topology optimization of material nonlinear continuum structures under stress constraints

This paper proposes a methodology to extend the bi-directional evolutionary structural optimization (BESO) method for compliance minimization (stiffness maximization) design of material nonlinear continuum structures subject to both constraints on volume fraction and maximum von Mises stress. BESO method based on discrete variables can effectively avoid the well-known stress singularity problem in density-based methods with low density elements. The aggregated p-norm global stress measure is adopted to approximate the maximum von Mises stress. The conventional compliance design objective is augmented with p-norm stress measures by introducing one Lagrange multiplier. The Lagrange multiplier is employed to yield compromised designs of compliance and the p-norm stress. A scheme is developed for the determination of the Lagrange multiplier so that the maximum von Mises stress could be effectively constrained. The update of the binary topological design variables lies in the sensitivity numbers derived using the adjoint method. As for the highly nonlinear stress behavior, both topological variables and sensitivity numbers are filtered to stabilize the optimization procedure. The filtered sensitivity numbers are further stabilized with their historical information. A series of comparison studies has been conducted to validate the effectiveness of the method on several benchmark design problems.

Bi-directional evolutionary stress-based topology optimization of material nonlinear structures

Stress-based topology optimization and nonlinear structural topology optimization is gaining increasing attention in order to make topology optimization more realistic. Thus, this paper extends current concepts of topology optimization to the design of structures made of nonlinear materials. An extended bi-directional evolutionary structural optimization (BESO) method for stress minimization topology optimization of material nonlinear structures is proposed in this work. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low-density elements. The maximum von Mises stress is approximated by the p-norm global stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topological variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several 2D benchmark design problems.

Adaptive multi-material topology optimization with hyperelastic materials under large deformations: A virtual element approach

We introduce a general multi-material topology optimization framework for large deformation problems that effectively handles an arbitrary number of candidate hyperelastic materials and addresses three major associated challenges: material interpolation, excessive distortion of low-density elements, and computational efficiency. To account for many nonlinear elastic materials, we propose a material interpolation scheme that, instead of interpolating multiple material parameters (such as Young’s modulus), interpolates multiple nonlinear stored-energy functions. To circumvent convergence difficulties caused by excessive distortions of low-density elements under large deformations, an energy interpolation scheme is revisited to account for multiple candidate hyperelastic materials. Computational efficiency is addressed from both structural analysis and optimization perspectives. To solve the nonlinear state equations efficiently, we employ the lower-order Virtual Element Method in conjunction with tailored adaptive mesh refinement and coarsening strategies. To efficiently update the design variables of the multi-material system, we exploit the separable nature and improve the ZPR (Zhang–Paulino–Ramos) update scheme to account for positive sensitivities and update the design variables associated with each volume constraint in parallel. Four design examples with three types of nonlinear material models demonstrate the efficiency and effectiveness of the proposed framework.

Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints

This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low density elements. The maximum von Mises stress is approximated by the p-norm global stress. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with p-norm stress. The stiffness and p-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topology variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several benchmark design problems.

An improved numerically-stable equivalent static loads (ESLs) algorithm based on energy-scaling ratio for stiffness topology optimization under crash loads

The standard equivalent static loads (ESLs) method for stiffness topology optimization under crash condition may lead to exaggerated equivalent loads, which is not appropriate to be incorporated into the linear static topology optimization and whereby hinder the optimization process. To overcome this disadvantage, an improved ESLs algorithm based on energy-scaling ratio is proposed to guarantee the numerical stability, especially for the first several cycles with relatively larger differences of strain energy between the original crash simulation and equivalent static analysis. At each cycle, the equivalent external static forces are calculated by multiplying the stiffness matrix and the displacement vector at the time with maximal strain energy during the crash simulation. A further adaptive energy-scaling operation for those forces are performed by a weighting factor of square root of the energy ratio to the standard equivalent static loads for the crash problem based on the judging criterion. The newly equivalent loads are incorporated into the static topology optimization, and topology results are filtered into a black-white design for the crash simulation to avoid the numerical issues due to existing of low-density elements. The process is repeated until the convergence criteria is satisfied. The effectiveness of the proposed method is demonstrated by investing two crash design problems.

A novel algorithm using an orthotropic material model for topology optimization

ABSTRACTThis article presents a novel algorithm for topology optimization using an orthotropic material model. Based on the virtual work principle, mathematical formulations for effective orthotropic material properties of an element containing two materials are derived. An algorithm is developed for structural topology optimization using four orthotropic material properties, instead of one density or area ratio, in each element as design variables. As an illustrative example, minimum compliance problems for linear and nonlinear structures are solved using the present algorithm in conjunction with the moving iso-surface threshold method. The present numerical results reveal that: (1) chequerboards and single-node connections are not present even without filtering; (2) final topologies do not contain large grey areas even using a unity penalty factor; and (3) the well-known numerical issues caused by low-density material when considering geometric nonlinearity are resolved by eliminating low-density elements in finite element analyses.

Elimination of the Effects of Low Density Elements in Topology Optimization of Buckling Structures

This paper presents an algorithm for structural topology optimization involving linear buckling. In this algorithm, finite element analysis (FEA) is conducted only in a domain with solid and gray elements, eliminating the contribution of low density elements; and the response function is constructed in the full design domain accounting the contribution of removed low density elements. The errors induced by removing void elements in FEA on eigenvalue and eigenvectors are analyzed. By introducing a dynamic low bound of the first eigenvalue and a load-path coefficient, the algorithm allows converged, nondisjointed and accurate solutions for topology optimization of structures involving buckling. Numerical results are presented for plate and column-beam structures against linear buckling to illustrate the efficiency and effectiveness of the present algorithm. Buckling experiments of the plates manufactured from the obtained topologies further verify the present algorithm.

Nonlinear response topology optimization using equivalent static loads—case studies

ABSTRACTNonlinear structural optimization is fairly expensive and difficult, because a large number of nonlinear analyses is required due to the large number of design variables involved in topology optimization. In element density based topology optimization, the low density elements create mesh distortion and the updating of finite element material with low density elements has a severe effect on the optimization results in the next cycles. In order to overcome these difficulties, the equivalent static loads method for nonlinear response structural optimization (ESLSO) primarily used for size and shape optimization has been applied to topology optimization. The nonlinear analysis is performed with the given loading conditions to calculate equivalent static loads (ESLs) and these ESLs are used to perform linear response optimization. In this paper, the authors have presented the results of five case studies with material, geometric and contact nonlinearities showing good agreement and providing justification of the proposed method.

Topology optimization of hyperelastic structures with frictionless contact supports

Hyperelastic structures usually undergo large deformations and thus may be subject to deformation-dependent contact supports. This paper presents an effective topology optimization methodology for the compliance-minimization design of hyperelastic structures with frictionless contact supports. In the optimization model, the strain-energy function of hyperelastic material is represented by an artificial penalization model, and the contact boundary conditions are modeled with hypothetical nonlinear springs. The additive hyperelasticity technique is employed for circumventing the local buckling instability exhibited by low-density elements. In conjunction with the adjoint variable sensitivity analysis, the nonlinear topology optimization problem is solved by a gradient-based mathematical programming algorithm. Numerical examples are given to show the importance of considering contact supports and to demonstrate the applicability of the proposed method.

Carbon distribution in multi-phase γ-TiAl based alloys and its influence on mechanical properties and phase formation

Advanced intermetallic γ-TiAl based alloys are attractive light-weight materials for high-temperature application. In order to extend their service temperature limits, alloying with low-density elements, such as C, is of particular interest and has been shown to effectively increase high-temperature strength as well as creep resistance.In the present study the local chemical composition of the constituent phases of the so-called TNM alloy and a C-containing derivative thereof is characterized by atom probe tomography. In both alloys Mo is found to preferentially locate in the βo phase, in contrast to Nb, which is dispersed in similar levels in all phases. In the C-containing alloy, C is enriched in the α2 phase, dissolved in the γ phase, but depleted in the βo phase. Furthermore, the investigation of interfaces through site-specific sample preparation reveals segregation of C at phase interfaces and their close vicinity. Finally, a correlation of the mechanical properties with the C distribution is established by nanoindentation technique. Both the γ and the α2 phase significantly harden through the addition of C, which is in good agreement with the C concentration present within these phases as observed by atom probe tomography. However, the βo phase softens through the addition of C, which is not a direct consequence of the C distribution, but follows from the absence of finely dispersed ωo particles in the βo phase of the C-containing alloy.

Topology optimization of continuum structure with dynamic constraints using mode identification

For the problems such as mode exchange and localized modes in topology optimization of continuum structure with dynamic constraints, it is difficult to apply the traditional optimization model which considers fixed order mode frequencies as constraints in optimization calculation. A new optimization model is established, in which the dynamical constraints are changed as frequencies of structural principal vibrations. The order of the principal vibrations is recognized through modal identification in the optimization process, and the constraints are updated to make the optimization calculation execute smoothly. Localized mode elimination techniques are introduced to reduce the localized modes induced by the low density elements, which could improve the optimization efficiency. A new optimization process is designed, which achieves the purpose of overcoming mode exchange problem and localized mode problem at the cost of increasing several structural analyses. Optimization system is developed by using Nastran to perform structural analysis and sensitivity analysis and two-level multipoint approximation algorithm as optimizer. Numerical results verified that the presented method is effective and reasonable.

Model dynamo may solve Mercury mystery

Simulations suggest that the planet’s top-heavy magnetic field derives from the unusual chemistry of its core.

Phase Stability of Low-Density, Multiprincipal Component Alloys Containing Aluminum, Magnesium, and Lithium

A series of low-density, multiprincipal component alloys containing high concentrations of Al, Mg, Li, Zn, Cu and/or Sn was designed using a strategy based on high-entropy alloys (HEAs). The alloys were prepared by induction melting under high-purity argon atmosphere, and the resulting microstructures were characterized in the as-cast condition. The resulting microstructures are multiphase and complex and contain significant volume fractions of disordered solutions and intermetallic compounds. By analyzing the atomic size difference, enthalpy of mixing, entropy of mixing, electronegativity difference, and valence electron concentration among the constituent elements, modified phase formation rules are developed for low-density multiprincipal component alloys that are more restrictive than previously established limits based on more frequently studied HEAs comprising mostly transition metals. It is concluded that disordered solid solution phases are generally less stable than competing ordered compounds when formulated from low-density elements including Al, Mg, and Li.

Modified ant colony optimization for topology optimization of geometrically nonlinear structures

Standard Ant Colony Optimization (ACO) algorithm cannot provide an optimal topology for geometrically nonlinear structural problems. The reason is that the tangential stiffness matrix of low-density elements may become zero or negative, which leads to serious numerical instability problems, or the standard ACO algorithm cannot search the really efficient elements because of improper element search method. A modified ant colony optimization (MACO) algorithm is suggested to improve computational efficiency and suitability of standard ACO algorithm in topology optimization for these problems. A continuous variable, called the “Element Contribution Significance (ECS),” is introduced, which serves to replace the positions of ants in the standard ACO algorithm, and assess the importance of each element in the optimization process. The optimized topologies based on MACO algorithm are compared to those of the other topology methods. From the comparison, it is verified that MACO algorithm can successfully be applied to topology optimization for geometrically nonlinear structures, as well as provide stable and robust topologies.

Towards the stabilization of the low density elements in topology optimization with large deformation

This work addresses the treatment of lower density regions of structures undergoing large deformations during the design process by the topology optimization method (TOM) based on the finite element method. During the design process the nonlinear elastic behavior of the structure is based on exact kinematics. The material model applied in the TOM is based on the solid isotropic microstructure with penalization approach. No void elements are deleted and all internal forces of the nodes surrounding the void elements are considered during the nonlinear equilibrium solution. The distribution of design variables is solved through the method of moving asymptotes, in which the sensitivity of the objective function is obtained directly. In addition, a continuation function and a nonlinear projection function are invoked to obtain a checkerboard free and mesh independent design. 2D examples with both plane strain and plane stress conditions hypothesis are presented and compared. The problem of instability is overcome by adopting a polyconvex constitutive model in conjunction with a suggested relaxation function to stabilize the excessive distorted elements. The exact tangent stiffness matrix is used. The optimal topology results are compared to the results obtained by using the classical Saint Venant---Kirchhoff constitutive law, and strong differences are found.