Multi-material topology optimization of laminated composite beam cross sections

The Gilbertown oil field is located along the southern boundary fault of the Gilbertown graben (Fig. 1). The Gilbertown graben trends east-west and is detached in the Jurassic Louann Salt at the updip limit of salt in the Gulf of Mexico basin (Pashin et al., 1998). The Gilbertown fault system forms the south side of the graben and the Melvin fault system forms the north side. The hydrocarbon production from the Cretaceous Selma chalk is from the rider blocks between both the East Gilbertown A-B and the West Gilbertown A-B fault segments (Fig. 1). Figure 1. Structure contour map of the top of the Selma Chalk, Gilbertown Field and vicinity (unsmoothed triangulation contouring), contour interval is 20 ft. The drag-style geometry in the rider block has been determined from regional 3-D subsurface mapping, area balancing (Pashin et al., 1998a, b), and two key dipmeter logs. The interpretation is illustrated (Fig. 2) by a cross section through a well (4195) which has a dipmeter log. The dipmeter shows hangingwall dips of 10† - 40† away from the master fault and horizontal footwall dips, indicating a drag fold confined to the hangingwall. Nearby wells have been projected onto the composite cross section according to their perpendicular distances from the East Gilbertown A fault at the top of the Selma in order to determine the probable extent of the oil production in the drag zone. Fractures and minor faults are interpreted from resistivity, FIL, and seismogram logs, caliper logs, and SCAT analysis (Bengtson, 1981) of the dipmeter logs. Fractures and minor faults are abundant in the drag zone in the hangingwall right above the master fault. Most secondary faults dip consistently north-northeast, synthetic and subparallel to the master fault, although they may diverge by up to 20† in strike. The associated drag fold axes are usually east-southeast trending and virtually horizontal. All hydrocarbons from the Selma chalk are produced from fractured regions within the drag zone. The proven productive area within the drag zone above the East Gilbertown A fault (Fig. 2) extends about at least ft vertically from the middle to the top of the Selma, and about 250 ft perpendicular to the master fault. End_Page 17------------------------ Figure 2. Composite cross section with drag fold interpretation in the vicinity of the master fault. Horizontal scale = vertical scale, A - G are faults interpreted from SCAT analysis. Wells are identified by their Alabama Oil and Gas Board permit numbers.

3-D beam element of composite cross section including warping and shear deformation effects

A new version of the Li\`ege intranuclear cascade (INC) model is proposed for the description of spallation reactions. Compared to the previous version, it incorporates new features: (i) it can accommodate a diffuse nuclear surface, (ii) the treatment of the Pauli blocking effect is improved, removing unphysical features linked with the use of statistical blocking factors, (iii) collisions between moving spectator nucleons are explicitly suppressed, (iv) pion dynamics is improved, especially concerning the delta lifetime, (v) it can accommodate light ions as incoming projecticles, (vi) the remnant angular momentum is included in the output of the model. Another important feature is the self-consistent determination of the stopping time, i.e., the time at which the INC calculation is terminated and coupled to evaporation. The predictions of the model, used with the Schmidt evaporation code, are tested against a large body of experimental data, in the 200-MeV--2-GeV range for incident energy per nucleon, including total reaction cross sections, neutron, proton, pion, and composite double differential cross sections, particle multiplicities, residue mass and charge distributions, and residue recoil velocity distributions. Good agreement is generally obtained without additional varying parameters. It is shown that the introduction of a diffuse surface considerably improves the description of the total reaction cross sections, of the intensity of the quasielastic peak in proton and neutron double differential cross sections and of the residue production yield for isotopes close to the target. High energy neutron spectra are found to be sensitive to details of the deuteron structure in deuteron-induced reactions. The shape of the fragmentation peaks in residue mass spectra is shown to be closely related to the shape of the distribution of the excitation energy left after the cascade stage. The longitudinal residue recoil velocity and its fluctuations display typical random-walk characterics, which are interpreted as a direct consequence of the independence of successive binary collisions occurring during the cascade process and therefore provide a strong support of the basic hypotheses of the INC model. Small but systematic discrepancies between model predictions and experiment are identified and possible further improvements to reduce them are discussed. The influence of the evaporation model is investigated. A comparison with similar approaches is presented.

In composite box girders with constant cross section corrugated steel webs, the shear stress in the web is typically calculated based on the applied shear force. For composite box girders with corrugated steel webs of variable cross sections, the calculation extends this approach by further incorporating additional shear effects induced by axial force and bending moment. The resulting web shear stress is obtained by superimposing the shear stress components arising from shear force, axial force, and bending moment. However, due to the unique folding effect of the corrugated steel web, its shear stiffness is significantly reduced. This effect also leads to discrepancies in the rotational angles among the top and bottom flanges, the web, and the overall cross section of the composite box girder. Although the traditional shear stress calculation method has clear physical meaning, it fails to accurately capture, through analytical expressions, the reduction in shear stiffness and the rotational angle discrepancies among these structural components. To address this, the present study derives a shear stress calculation equation for composite box girders with corrugated steel webs, accounting for the shear stiffness reduction caused by the folding effect. A rotational angle displacement function is introduced to reflect the angular discrepancies among the top and bottom flanges, the web, and the composite cross section. The web shear stress is then determined by combining the effects of shear stiffness reduction and rotational angle variation. Using the energy variational method, the shear contributions of the flanges and the web are decoupled. The shear force carried by the flanges is further decomposed into contributions from the top and bottom plates based on moment equivalence. In addition, the shear force distribution ratios among the top plate, bottom plate, and web are examined. Finally, a cantilever beam example is presented to validate the proposed equation by comparing it with existing shear stress calculation expressions for corrugated steel webs, thereby evaluating the shear performance of composite box girders.

<div class="section abstract"><div class="htmlview paragraph">Evolving fuel efficiency and emissions standards, along with consumer demand for performance, are strong pressures for light-weighting of performance oriented motorcycles. The field of topology optimization (TO), with the extension of multi-material topology optimization (MMTO) provide manufacturers with advanced structural light-weighting methodology. TO methodology has been adopted in many industries, including automotive where light-weighting assists in meeting efficiency regulations. The development of process specific manufacturing constraints within MMTO is a critical step in increasing adoption within industries dealing with manufacturing cost restrictions. This capability can decrease design complexity, lowering manufacturing costs of optimization solutions.</div><div class="htmlview paragraph">A conventional all-aluminum perimeter style motorcycle chassis is analyzed to develop baseline compliance (total strain energy) metrics. An MMTO design space is created and optimized with steel and aluminum, such that results match the baseline design weight. This formulation demonstrates increased structural efficiency through stiffer structures at equivalent weight. Results are generated with standard MMTO, symmetry, and extrusion constraints to demonstrate utility of these manufacturing constraints. Material ratios are used to enforce lower cost material distribution selections.</div><div class="htmlview paragraph">The usage of MMTO with manufacturing and material ratio constraints has resulted in up to 60.4% reduction in structural compliance of designs, and a multitude of lower cost alternative designs. The usage of symmetry constraints provides effectively identical results to standard MMTO, with a computational time reduction of 29%. Extrusion constraints demonstrate decreased manufacturing difficulty with a computational time reduction of 52% and structural performance penalty of 58.6% from the lowest compliance MMTO result. The enforcement of steel has demonstrated a decrease in structural performance (increase in compliance) and material costs, with varying degrees depending on manufacturing constraints and enforcement limits. The MMTO based designs provide a range of solutions to designers, which can be selected based on the importance of structural efficiency, manufacturing difficulty, and material costs.</div></div>

Microstructure design of porous viscoelastic composites with prescribed relaxation moduli by multi-material topology optimization

ABSTRACTA comprehensive solution for bus frame design is proposed to bridge multi-material topology optimization and cross-sectional size optimization. Three types of variables (material, topology and size) and two types of constraints (static stiffness and frequencies) are considered to promote this practical design. For multi-material topology optimization, an ordered solid isotropic material with penalization interpolation is used to transform the multi-material selection problem into a pure topology optimization problem, without introducing new design variables. Then, based on the previously optimal topology result, cross-sectional sizes of the bus frame are optimized to further seek the least mass. Sequential linear programming is preferred to solve the two structural optimization problems. Finally, an engineering example verifies the effectiveness of the presented method, which bridges the gap between topology optimization and size optimization, and achieves a more lightweight bus frame than traditional single-material topology optimization.

This paper deals with simultaneous topology and shape optimization of elastic contact problems. The structural optimization problem for an elastic contact problem is formulated. Shape as well as topological derivatives formulae of the cost functional are provided using material derivative and asymptotic expansion methods, respectively. These derivatives are employed to formulate necessary optimality condition for simultaneous shape and topology optimization and to calculate a descent direction in numerical algorithm. Level set based numerical algorithm for the solution of this optimization problem is proposed. Numerical examples are provided and discussed.

According to the requirements of the aerospace industry for high strength, high stiffness, and lightweight structural parts, topology optimization has been proved to be an effective product design method. As one of the most conceptual and prospective structural optimization design methods, topology optimization intends to seek the optimal layout of materials in an allowed design region under a given load and boundary conditions. Thus, the object of study in the article is the method of topological optimization of aircraft structures. The goal of this article is to analyze the existing approaches, algorithms, as well as application of the method of topological optimization in the aerospace field in applied problems. The tasks are to describe the existing various approaches methods, features, and research directions of topological optimization as well as to study the possibility of application in the manufacturing process of composite structures. The following results were obtained. The optimization methods are briefly explained and compared, and the advantages and limitations of each approach are discussed. The various ways of simultaneous optimization of fiber orientation and structural topology were described and analyzed. The features of different methods of continuous fiber orientation optimization method were reviewed. The discrete fiber orientation optimization methods were represented. The possibility of multi-scale concurrent topological optimization was described. The combination of topology optimization and additive manufacturing was considered. Finally, the topology optimization of FRC structures which have been resolved in literature are reviewed and the potential research fields requiring more investigation are pointed out. Conclusions. In the article, a comprehensive review of the topology optimization design of FRC structures was presented. The promising way is to combine topology optimization with additive manufacturing techniques. However, these proposed methods may not suitable for other more complex problems, such as bucking stability and natural frequency. Hence, the topology optimization design of complex FRC components under complicated conditions is the main challenge in the future. This can be a new trend in the topology design of FRC structures.

The algorithms for multi-material topology optimization were developed to solve compliance-minimization problems and applied to engineering problems in automotive concepts and lightweight design. Two small-scale problems of a long cantilever and a control arm were studied initially to verify the effectiveness of the developed algorithms and in-house program. Optimal solutions achieved by the multi-material topology optimization method developed were compared to their counterparts obtained by standard single-material topology optimization. To efficiently solve real-world engineering problems, the algorithms were further advanced to incorporate extrusion constraints and to handle multiple load cases. The effectiveness and the efficiency of the proposed method were demonstrated by the study of two real-world engineering problems: (a) the conceptual design of a cross-member for a chassis frame; and (b) the conceptual design of an automotive engine cradle. The two optimization design problems both involved complex geometries, design and non-design domains, prescribed regions with specific material allocations, multiple load cases, and manufacturing extrusion constraints. It was explicitly demonstrated that, for the same weight, the optimum designs achieved by the multi-material topology optimization method were stiffer than those achieved by standard single-material topology optimization.

Subcommittee 20—Composite Columns was designated in 1973 as a standing committee of the Structural Stability Research Council (formerly called the Column Research Council). With an abundant background of experience regarding steel column behavior, the Council recognized that steel-concrete composite compression members should behave almost the same as plain steel columns if, in composite cross sections, the strength and stiffness of the structural steel alone were several times greater than the strength and stiffness of the structural concrete. The Council was also aware that if, in a composite cross section, the strength and stiffness of the concrete alone were significantly greater than the strength and stiffness of structural steel, the composite compression member would behave much the same as an ordinary reinforced concrete column. Design concepts traditionally applied to structural steel involved fundamental differences from those generally applied to reinforced concrete. The consequences of unequal results from the different design concepts required reconciliation within a rational statement of recommended practice for composite column design.

In this paper we consider the classical minimum compliance topology optimization in linear elasticity. In our approach, the objective is to minimize a cost functional that comprises the work of applied body and traction forces obtaining simultaneously the optimal topology and the optimal localization for proportional actuators on the concerning structural domain. A proportional actuator is considered as a spring and one of the objectives is to obtain the optimal local to couple it on the structure. The problem is treated as a topology optimization problem where a given cost functional must be minimized under suitable constraints for external forces, control energy, volume restriction and a set of project variables. The topology optimization procedure here performed uses the solid isotropic material with penalization approach, based on the concept of optimizing the material distribution. The solution of the optimization problem is obtained by using a sequential linear programming and the finite element method is used to discretize the design domain. Several numerical examples are presented showing the effectiveness of the proposed approach.

This paper focuses on simultaneous topology and fiber layout optimization for additive manufacturing processes. Several manufacturing limitations stand in the way of manufacturing highly efficient composite parts with additive manufacturing. Further, the use of post-processing in order to fulfill manufacturing limitations may reduce the performance of the optimized result to an unknown degree. In addition, the post-processed results may not satisfy structural constraints. This paper utilizes the Moving Morphable Components framework for topology optimization. Each component is parametrized by a skeleton curve defined based on the Absolute Nodal Coordinate Formulation. This formulation allows fibers to be defined parallel to the skeleton curve within a component using a novel projection method. Further, this method enables the definition of novel manufacturing constraints on the parametric component. To achieve inter-component fiber continuity, without a priori knowledge of the optimized component layout, a novel multi-objective formulation is presented. Lastly, a failure index constraint based on the Tsai–Wu failure criterion is added to the formulation. The results show that the optimized results can be manufactured with very limited post-processing required.

This paper presents a new strategy to distribute two different materials for multi-material topology optimization. Extended from the bi-directional evolutionary structural optimization (BESO) method for a single material, the multi-material bidirectional evolutionary structural optimization (MBESO) method has been developed, which can effectively handle the topology optimization problems involving two materials like steel and concrete. However, in some special cases, overloading of part of the compressed material occurs in the multi-material structures designed by using the MBESO method. Aimed to solve this critical problem, a simple but effective strategy is proposed in this paper. In steel-concrete composite structures, for instance, the overloaded compressed concrete elements with exceedingly high stress are replaced with steel material. The small amount of steel material added to the highly compressed region can effectively reduce the maximum compressive stress of the concrete material to a safe level. The comparison between the original MBESO method and the improved strategy based on a series of two-dimensional and three-dimensional examples clearly demonstrates the effectiveness of the proposed strategy in enhancing the structural safety and strength of the topologically optimized composite structures. This distinctly different material distribution strategy shows its potential and value in multi-material topology optimization research and applications.

A new method of simultaneous optimization of geometry and topology is presented for plane and spatial trusses. Compliance under single loading condition is minimized for specified structural volume. The difficulties due to existence of melting nodes are successfully avoided by considering force density, which is the ratio of axial force to the member length, as design variable. By using the fact that the optimal truss is statically determinate with the same absolute value of stress in existing members, the compliance and structural volume are expressed as explicit functions of force density only. After obtaining optimal cross-sectional area, nodal locations, and topology, the cross-sectional areas and nodal coordinates are further optimized using a conventional method of nonlinear programming. Accuracy of the optimal solution is verified through examples of plane trusses and a spatial truss. It is shown that various nearly optimal solutions can be found using the proposed method.