Linear Static Analysis

Structural analysis aims to predict deformation and stresses in a body (or collection of bodies) that is restrained from moving and is subjected to external forces (loads). Different theoretical models of structural analysis have been developed to simulate a variety of realistic physical behaviors and phenomena. The simplest and the most widely used type of structural analysis is linear static analysis, based on theory of lineary elasticity.

Like all numerical simulations methods, Scan&Solve™ numerically approximates this idealized model of linear elasticity within some limited precision. As such, Scan&Solve™ can provide insight into structural properties of solid shapes and help in choosing the best available alternative. However, numerical simulation of an idealized mathematical model is not a substitute for physical testing and should not be relied upon for critical design decisions.

• Linear Elasticity
• Scan&Solve Numerical Model
• Physical Reality
• Known Limitations

Scan&Solve™ simulates linear static behavior of 3D solids based on mathematical theory of linear elasticity which approximates physical reality in many common situations. Like all mathematical models, linear elasticity idealizes physical reality, making a number of simplifying assumptions.

Static: This assumption neglects all dynamic (time-varying) forces and amounts to assuming that all loads are increased slowly to the specified magnitudes, and then remain constant.

Elastic: No permanent deformation takes place, and the body returns to its original shape if the loads are removed.

Linearity: Model deformations (displacements) are linearly proportional to applied loads (forces). For example, doubling the magnitude of the force will double the magnitude of the resulting deformations.

Linear static analysis predicts the magnitude of stresses and elastic displacements within the body. In locations where the magnitude of stresses exceed certain levels, linear static analysis predicts material failure based on several experimentally verified failure criteria. The type of failure depends on the type of material and the stress levels; linear static analysis cannot predict whether failure results in large permanent deformation, cracks, or breakage, but only that the stresses and displacements will exceed the elastic limit of the material.

Scan&Solve™ is a generalized form of the classical Finite Element Analysis (FEA). Classical FEA approximates an idealized theoretical model of physical behavior by breaking up the model or space into small pieces called finite elements. But Scan&Solve™ was developed specifically to liberate FEA from the tyranny of meshing, while preserving most of the advantages of this classical and widely accepted method of engineering analysis. The basic idea is simple: create separate geometric and physical representations of the model in question and combine them only when necessary, without requiring expensive and error-prone data conversions and always using the most authentic representation available. The concept is illustrated below.

The analysis model is constructed on a (typically, but not necessarily) uniform orthogonal grid of space that initially knows nothing about the model being analyzed. It can be thought of as a 3D “graph paper”. The usual variety of basis functions may be associated with the vertices of this mesh. Currently, only linear B-spline functions are released. The geometric model exists in the same space, in its native unaltered form, and is not aware of the mesh surrounding it. The usual FEA procedure is then modified at run time to account for the existence of the geometric boundaries, restraints, and loads via the Scan&Solve™ process that eliminates or modifies the relevant finite elements. From the user perspective, the whole process is totally transparent and mesh-free, or more accurately “meshing-free”.

Here is the comparison of the classical mesh-based FEA and Scan&Solve™ technology. If you want to know more technical details about the Scan&Solve™ technology, you can find them in this white paper.

It is important to remember that Scan&Solve™ computes a numerical approximation of an idealized theoretical model, not physical reality. Every model has its limitations. For example, the linear theory of elasticity predicts infinite stresses near “wedges”, re-entrant corners, interfaces between different materials, and so on. In physical reality, this cannot happen, because the material simply deforms more “plastically” (as opposed to “elastically”). But in the computer simulation, this means that at some points in your model, stresses will never converge – they will just get bigger and bigger as you increase the resolution. The more complex your model is, the more likely you will have some points like that.

More generally, the linear static model of elasticity (and hence Scan&Solve™) does not account for many important physical phenomena, including vibration, buckling, material and geometry non-linearities, large and plastic deformations, and so on.

Scan&Solve computes a numerical approximation of a mathematical model of linear elasticity. As is the case with all such software, the computed stresses and displacements should not be confused with physical reality and should not be used for critical design decisions in place of physical experiments. A number of factors, briefly described below, can cause Scan&Solve answers to deviate significantly from physical reality:

• Model is not linear elastic. A linear elastic model of stress is not applicable in many situations, e.g. non-linear material properties, large plastic deformations, vibrations, buckling, thermal stresses, and so on.
• Idealized geometry and material properties. While geometric inaccuracies may be quantified by design and manufacturing tolerances, predicting the effect of material variability is usually more difficult.
• Imprecise or unrealistic boundary conditions. The correctness of the model depends critically on the type (are they really static?) and accuracy of restraints and loads. But these parameters are rarely known precisely, and are sometimes simplified or idealized, contributing to uncertainty of the computed results. For example, edge restraints are non-physical idealizations of small-face restraints; their use may result in poor solutions in the vicinity of such edges.
• Low resolution. Every known stress analysis approach (including Scan&Solve) relies on spatial discretization of the geometry in one way or another. Scan&Solve is unique because it uses volumetric discretization of space that does not conform to geometry, and the user does not need to deal with meshing, but fundamentally, it still relies on the notion of a “finite element” which is associated with some volumetric portion of geometry. It is usually impossible to estimate a priori how many elements are sufficient for accurate answers. Scan&Solve currently does not perform any posteriori analysis of the computed solutions for accuracy or convergence.
• Missed small features. Scan&Solve tolerates small imperfections and features in the noisy or overly detailed boundary representation of a solid. However, this means that at low resolution, it could also miss some important features, such as small holes, gaps, and channels that could affect accuracy or correctness of computed solutions.
• Large size to scale ratio. The size to scale ratio is a fundamental barrier of all numerical methods, and not a limitation unique to Scan&Solve. Challenging models include thin-walled solids, slender truss-like structures, solids restrained on very small faces, and other models where the ratio of the overall size of the solid to the smallest dimension ('scale') on the object is very large. Accurate analysis requires that the size of finite element should be small enough to resolve the smallest scale of the mode. But if the model is also very large, this will result in a huge number of elements, easily exceeding the maximum resolution available in Scan&Solve.
• No adaptive analysis. Some of the above problem may be alleviated by varying the size of the elements throughout the space, depending on the size of small features and/or desired resolution. The current version of Scan&Solve does not support such adaptive analysis.
• No point restraints. Only surface areas and edges can be restrained in the current version of Scan&Solve, as dictated by the theory. Point restraints are commonly used in mesh-based FEA packages where they are applied to nodes. But such restraints are actually not physical and may result in poor solutions in the vicinity of the restraints. Although edges can be restrained, please remember that they are also non-physical idealizations of small-face restraints. Using edge restraints may result in poor solutions in the vicinities of the restrained edges.
• First degree numerical approximation. The current version of Scan&Solve relies on first degree (linear) polynomial functions (splines) to approximate the solution. Some structural analysis problems require higher-degree functions to achieve accuracy and convergence. We expect future versions of Scan&Solve to support such functions.