Topological design of modular structures under arbitrary loading

Crack energy density of a piezoelectric material under general electromechanical loading

This paper assesses the applicability of bearing capacity theory for evaluating the forces generated on wheels operating on clay under steady rolling conditions. Considering advances in bearing capacity theory, in particular the interaction diagrams developed for general loading, a theoretical model for computing the horizontal force or torque from fundamental input parameters such as the vertical force (weight), wheel diameter, and undrained shear strength of the soil is presented. The predictions are compared with existing analytical solutions and data from laboratory testing, and reasonable agreement is demonstrated. The newly proposed model provides a means to predict wheel forces analytically under any operating condition (driven, braked, or towed), provided the contact length and so-called contact angle, which defines the position of the contact interface, can be estimated. The model provides a rigorous, convenient framework for evaluating wheel forces under arbitrary loading and enables a natural physical interpretation of the mobility problem.

The problem of a circular plate, with a central hole, elastically constrained against rotation and deflection, acted upon by a transverse linear temperature gradient or a general axisymmetrical loading condition is considered in this paper. With the aid of presented graphs and simple algebraic equations, it is a relatively simple matter to construct the desired deflection and stress pattern for any combination of elastic constraints and load distribution.

The mechanical behaviour of tumour and non-tumour cells under vibration remains insufficiently explored, particularly the role of elastic modulus in dynamic responses. This study investigates the vibration-induced mechanical behaviour in cellular structures with varying elastic moduli (E = 0.1 , 1 , and 10 kPa) and aspect ratios ( and 4), focusing on vertical and horizontal forced vibrations. Finite element analysis was conducted to evaluate the natural frequencies, mode shapes, membrane accelerations, and stress responses. The intermediate aspect-ratio structures () exhibited higher natural frequencies but a 64.2% increase in stress concentration, making them more susceptible to localised deformation under resonance. Conversely, higher aspect-ratio structures () demonstrated improved vibrational stability with reduced resonance peaks and 64.6% lower localised stress. This study further confirmed that vertical vibrations generate higher stress and acceleration than horizontal vibrations owing to gravitational effects. Stress contour analysis indicated that under low-intensity vibrations, intermediate aspect-ratio structures may exceed their yield stress thresholds, leading to potential membrane rupture. These findings suggest that vibration-induced mechanical stimulation can be sensed differently depending on the elastic moduli and aspect ratios of tumour and non-tumour cells.

New, exact nonlinear virtual work equations for plane frames are rigorously derived and applied to problems of ‘frames’, whose members maybe treated as inextensible, as opposed to ‘trusses’. The virtual work equations are used to establish a simple stability discriminant (non-negative for stability); to investigate the orthogonality and completeness properties of the sequences of eigenfunction displacement modes generated by arbitrary axial loading; and to discuss various features of the response to non-axial loading. The concept of the ‘ current critical load’, which determines the stability of the frame, is introduced, and certain principles governing bifurcation under non-axial loading are established. Two methods of computing the nonlinear equilibrium path of an arbitrary elastic frame under general loading are developed, and an example is given of the direct evaluation of the maximum load according to small displacement theory.

This paper presents an approach to reframe the sizing problem for vertical-lift unmanned aerial vehicles (UAVs) as an optimization problem and obtains a weight-optimal solution with up to two orders of magnitude of savings in wall clock time. Because sizing is performed with higher fidelity models and design variables from several disciplines, the Simultaneous Analysis aNd Design (SAND) approach from fixed-wing multidisciplinary optimization literature is adapted for the UAV sizing task. Governing equations and disciplinary design variables that are usually self-contained within disciplines (airframe tube sizes, trim variables, and trim equations) are migrated to the sizing optimizer and added as design variables and (in)equality constraints. For sizing consistency, the iterative weight convergence loop is replaced by a coupling variable and associated equality consistency constraint for the sizing optimizer. Cruise airspeed is also added as a design variable and driven by the sizing optimizer. The methodology is demonstrated for sizing a package delivery vehicle (a lift-augment quadrotor biplane tailsitter) with up to 39 design variables and 201 constraints. Gradient-based optimizations were initiated from different starting points; without blade shape design in sizing, all processes converged to the same minimum, indicating that the design space is convex for the chosen bounds, constraints, and objective function. Several optimization schemes were investigated by moving combinations of relevant disciplines (airframe sizing with finite element analysis, vehicle trim, and blade aerodynamic shape design) to the sizing optimizer. The biggest advantage of the SAND strategy is its scope for parallelization, and the inherent ability to drive the design away from regions where disciplinary analyses (e. g., trim) cannot find a solution, obviating the need for ad hoc penalty functions. Even in serial mode, the SAND optimization strategy yields results in the shortest wall clock time compared to all other approaches.

The multigrid method in the form of MGCS V-cycles with matrix-dependent prolongations is implemented for solving linear equations resulting from a standard discretization of a 2D elliptic equation (5-point “cross” stencil). Simplicity of input data allows incorporation of the solver into any relevant commercial or research code. The performance of MGCS is compared against other iterative (in-house BiCGStab preconditioned by ILU(2) factorization, PyAMG) and direct (DSS from Intel MKL library) solvers. The following performance tests are considered: 1) synthetically-generated matrices with large contrast in elements; 2) matrices formed during simulations of multiphase flows in hydraulic fractures/slots with large viscosity contrast (up to 106). Dependence of the performance of MGCS on the number of smoothing iterations at each mesh level and aspect ratio of mesh cells is studied. It is found that the best performance is gained by MGCS V-cycles with a single smoothing iteration per mesh level on meshes with square cells. Increasing the aspect ratio of the mesh cells and increasing the number of smoothing iterations per mesh level typically slows down the convergence rate. The MGCS code shows the best performance as compared to other solvers tested. The speed-up increases significantly with an increase in the mesh resolution.

The aim of this study is to deal with the simulation of intralaminar fatigue damage in unidirectional composite under multi-axial and variable amplitude loadings. The variable amplitude and multi-axial loading is accounted for by using the damage hysteresis operator based on Brokate method [6]. The proposed damage model for fatigue is based on stiffness degradation laws from Van Paepegem combined with the ‘damage’ cycle jump approach extended to deal with unidirectional carbon fibres. The parameter identification method is here presented and parameter sensitivities are discussed. The initial static damage of the material is accounted for by using the Ladevèze damage model and the permanent shear strain accumulation based on Van Paepegem’s formulation. This approach is implemented into commercial software (Siemens PLM). The validation case is run on a bending test coupon (with arbitrary stacking sequence and load level) in order to minimise the risk of inter-laminar damages. This intra-laminar fatigue damage model combined efficient methods with a low number of tests to identify the parameters of the stiffness degradation law, this overall procedure for fatigue life prediction is demonstrated to be cost efficient at industrial level. This work concludes on the next challenges to be addressed (validation tests, multiple-loadings validation, failure criteria, inter-laminar damages…).

2D weight function development using a complex Taylor series expansion method

A newly developed multifrequency wireless power transfer system is designed and presented in this article. Using multiple frequency channels, the system can simultaneously deliver power to different wireless pickups with arbitrary power requirements or load characteristics. The proposed design scheme of the system topology realizes both stable power flow control and precise power adaption. The load-independent constant current output has been successfully achieved under multifrequency operation mode, which increases both system stability and fault-tolerance ability. The output characteristic can also immunize the system against undesired interferences from the reflected impedance of different receivers at other power transmission channels. Both finite element analysis and circuit simulations have verified good performance of the system under different power adaption requirements. An experimental prototype can successfully deliver a total of 18 W wireless power simultaneously to two identical resistive loads at the desired power requirement distribution of 9:1 via the same transmitter. It shows good performance in both stability and controllability.

The goal of this study is to evaluate and design steel plates with optimal material distributions achieved through a specific material topology optimization by using a CCARAT (Computer Aided Research Analysis Tool) as an optimizer, topologically optimally updating node densities as design variables. In typical material topology optimization, optimal topology and layouts are described by distributing element densities (from almost 0 to 1), which are arithmetic means of node densities. The average element densities are employed as material properties of each element in finite element analysis. CCARAT may deal with material topology optimization to address the mean compliance problem of structural mechanical problems. This consists of three computational steps: finite element analysis, sensitivity analysis, and optimality criteria optimizer updating node densities. The present node density based design via CCARAT using node densities as design variables removes jagged optimal layouts and checkerboard patterns, which are disadvantages of classical material topology optimization using element densities as design variables. Numerical applications that topologically optimize reinforcement material distribution of steel plates of a cantilever type are studied to verify the numerical superiority of the present node density based design via CCARAT.

As advanced manufacturing techniques such as additive manufacturing become widely available, it is of interest to investigate the potential advantages that arise when designing periodic metamaterials to achieve a specific desired behavior or physical property. Designing the fine scale detailed geometry of periodic metamaterials to achieve a specified behavior falls under the category of notoriously intractable inverse problems. To simplify solving the inverse problem, most relevant works represent metamaterials as periodic single unit cell structures repeated in regular lattices. Such representation simplifies modeling and simulation task but at the cost of possibly limiting the range of physical behaviors that can be achieved through the use of more than one unit cell structures. This article outlines a quasi-periodic representation that utilizes more than a single unit cell to generate periodic metamaterials. Additionally, a hierarchical optimization scheme to optimize the generating function for a quasi-periodic structure using the genetic algorithm (GA) and a barrier function interior point method is also sketched to solve the inverse problem. To demonstrate the utility of the proposed hierarchical optimization framework to solve quasi-periodic metamaterial inverse problem, a problem in which the objective is to minimize the total strain in the structure while subjected to weight and the total-size constraint is considered. We detail the overall computational approach in which geometric representation, optimization algorithms, and finite element analysis are coupled and report preliminary numerical experiments.

Dimension reduction and surrogate based topology optimization of periodic structures

Atmospheric conditions in which cellular-roll cloud patterns are apparent, and dependence of the aspect ratios of the cells on depths of the convective layer and the stability ratio are investigated by examining satellite pictures and aerological data over eastern Asia during cold air outbreaks in the months January to March for 1975–82. Vertical profiles of temperature and relative humidity in the convective layer along the cloud bands over the ocean show an increase of static stability, relative humidity, and layer depth in the downwind direction. Longitudinal convective rolls are realized when vertical wind shear is between 10−3 and 10−2 sec−1, and three-dimensional open cells are realized below these values. The cloud top height is proportional to l3/5 of the cloud-band spacing for l≲20 km and to l2/5 for l≳20 km. The obtained aspect ratio for longitudinal rolls ranges from 5 to 18, increasing in the downwind direction at a rate of 1/110 km, while open cells are obtained in the range from 8 t...

In the finite element modelling of structural frames, external loads such as wind loads, dead loads and imposed loads usually act along the elements rather than at the nodes only. Conventionally, when an element is subjected to these general transverse element loads, they are usually converted to nodal forces acting at the ends of the elements by either lumping or consistent load approaches. In addition, it is especially important for an element subjected to the first- and second-order elastic behaviour, to which the steel structure is critically prone to; in particular the thin-walled steel structures, when the stocky element section may be generally critical to the inelastic behaviour. In this sense, the accurate first- and second-order elastic displacement solutions of element load effect along an element is vitally crucial, but cannot be simulated using neither numerical nodal nor consistent load methods alone, as long as no equilibrium condition is enforced in the finite element formulation, which can inevitably impair the structural safety of the steel structure particularly. It can be therefore regarded as a unique element load method to account for the element load nonlinearly. If accurate displacement solution is targeted for simulating the first- and second-order elastic behaviour on an element on the basis of sophisticated non-linear element stiffness formulation, the numerous prescribed stiffness matrices must indispensably be used for the plethora of specific transverse element loading patterns encountered. In order to circumvent this shortcoming, the present paper proposes a numerical technique to include the transverse element loading in the non-linear stiffness formulation without numerous prescribed stiffness matrices, and which is able to predict structural responses involving the effect of first-order element loads as well as the second-order coupling effect between the transverse load and axial force in the element. This paper shows that the principle of superposition can be applied to derive the generalized stiffness formulation for element load effect, so that the form of the stiffness matrix remains unchanged with respect to the specific loading patterns, but with only the magnitude of the loading (element load coefficients) being needed to be adjusted in the stiffness formulation, and subsequently the non-linear effect on element loadings can be commensurate by updating the magnitude of element load coefficients through the non-linear solution procedures. In principle, the element loading distribution is converted into a single loading magnitude at mid-span in order to provide the initial perturbation for triggering the member bowing effect due to its transverse element loads. This approach in turn sacrifices the effect of element loading distribution except at mid-span. Therefore, it can be foreseen that the load-deflection behaviour may not be as accurate as those at mid-span, but its discrepancy is still trivial as proved. This novelty allows for a very useful generalised stiffness formulation for a single higher-order element with arbitrary transverse loading patterns to be formulated. Moreover, another significance of this paper is placed on shifting the nodal response (system analysis) to both nodal and element response (sophisticated element formulation). For the conventional finite element method, such as the cubic element, all accurate solutions can be only found at node. It means no accurate and reliable structural safety can be ensured within an element, and as a result, it hinders the engineering applications. The results of the paper are verified using analytical stability function studies, as well as with numerical results reported by independent researchers on several simple frames.