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Another use for geometric constraints and 3D sketches

Dr. Jody Muelaner, PhD CEng MIMechE

I recently wrote about some of the less well-known uses for sketches in parametric CAD software such as SolidWorks. The geometric constraint solvers that the sketch function is based on does some really clever optimization to solve very complex geometry problems in a highly intuitive graphical way. This functionality is included so that model dimensions can be easily changed and part geometry then automatically updated, preserving design intent. However, the geometry solving capability can be repurposed in many ways to create extremely powerful and intuitive models and optimizations. My previous article looked at how geometric constraints can be used to optimize linkage lengths and pivot positions to synthesize mechanisms that produce specified types of motion. In this article, I detail a case study where 3D sketches in SolidWorks were used to validate a complete ray path simulation being used to design an interferometer in MATLAB.

The Interferometer Design
The simulation was for a new laser measurement technique that would allow highly accurate and dynamic measurements between steel spheres. Steel spheres can be manufactured with a very accurately defined radius, allowing measurements from multiple directions to reference a common point at the center of the sphere. It would therefore be possible to measure multiple 3D coordinates with very high accuracy and full traceability to fundamental length standards. The novel interferometer which allows these measurements has been designed by researchers at the University of Oxford, University of Bath and the National Physical Laboratory (NPL).

The technique, known as Absolute Multilateration between Spheres (AMS), could one day allow the measurement of coordinates to within a few micrometers, over distances of tens of meters and with no environmental controls. Because the laser path is contained in a tube, environmental parameters can be controlled and measured for the beam path without controlling the entire environment. This could enable the next generation of aerospace structures to achieve natural laminar flow and part-to-part interchangeability.

The diagram below shows the interferometer arrangement which allows an absolute distance measurement between steel spheres. It uses a fiber channeled laser source which can be shared between many individual distance measurements. The laser passes through a number of wave plates, changing its polarization plane and therefore allowing a polarization-dependent beam splitter to either reflect or pass the beam, depending on where it is in its path. The measurement beam, therefore, goes from the splitter to a collimating lens A, it then reflects off the beam splitter at B, reflects off the first sphere C, passes through the beam splitter, reflects off the second sphere D, reflects off the beam splitter at E and finally hits the detector at F. At the same time, a reference beam goes straight from A to F.

Due to the convex nature of the spheres, the laser beam diverges when it is reflected, fanning out the rays into a cone shape. This means that only a small fraction of the measurement beam will reach the detector.

The MATLAB Simulation
A MATLAB simulation was developed which included a mathematical function for the path of a single ray ABCDEF. This function used vector geometry and had variables to represent the position of the ray in the beam at A and for the alignments of the components in the system. A numerical integration was then carried out which simulated a number of discrete rays within the beam to determine the laser power at the detector, the error of distance measurement and the ability to clearly resolve fringes.

With alignment errors, the path ABCDEF becomes A’B’C’D’E’F’. A ray on the reference path travels from point Ar to interfere with the measurement path ray at point F’. The coordinate system was centered at point B with the x-axis on path BA and the y-axis on path BC.

Vector geometry can be used to find each of the points in the path in turn. For example, the direction of vector A’B’ is given by applying rotation matrices to the nominal direction vector:

The intersection of this vector with the plane of the beam splitter gives the position of B and the direction of B’C’ can be found by reflecting A’B’ about a vector normal to the beam splitter (N_x):

The intersection of the first sphere with vector B’C’ gives two possible positions for C. The one with the minimum y-coordinate is the correct one. The direction of C’D’ is then found by reflecting B’C’ about the surface normal to the sphere at point C’. The surface normal has the direction from the center of the sphere to the point C’. A similar process is carried out to model the rest of the ray path.

Because of the highly divergent nature of the beam, it was necessary to only simulate the very small part which would actually reach the detector. Simulating the entire beam with sufficient resolution to give meaningful results at the detector would have taken many years of computational solve time. However, due to alignment errors, it was not possible to determine which part of the beam would reach the detector without simulating the ray path. Therefore, an initial search had to be carried out to find the edges of the detector. These capabilities are not available in standard ray tracing software and therefore a bespoke MATLAB simulation was required.

Validating the Model in SolidWorks
A vector geometry model for a ray path with nominal geometry was relatively straight forward. However, when the alignment errors of each component were also included the model became somewhat complex and very difficult to directly verify. SolidWorks was used to validate the ray path model. A geometric constraint-based sketch provided a really useful and intuitive way to do this.

The first stage in creating the SolidWorks validation model was to create geometric elements for the spheres, beam splitter, laser source and detector. The spheres were modeled as solids, the beam splitter and detector were modeled as reference planes.

The next stage was to model the actual ray path. Each straight line section of the ray path was modeled using a 3D sketch containing a line to represent the path. The surface normal for each surface of reflection was also included in the sketch as another straight line.

The first straight-line section of the ray path is AB. It originates from and in the direction of the laser source. Dimensions were included to represent the alignment errors. The endpoint of this line is set to be coincident with the beam splitter plane to find the position of point B.

The reflection of AB on the beam splitter was then modeled by finding the surface normal direction and reflecting on this vector. Within SolidWorks, the surface normal vector was modeled as an axis which is coincident with the endpoint of line AB and which is perpendicular to the plane of the beam splitter. Next, a reference plane was created for the plane of reflection, this is coincident with both the surface normal vector and the other endpoint of the line AB, at point A. Sketching on this plane, the direction vector for BC is a line that is symmetrical with the line AB about a centerline along the surface normal axis.

Once the direction vector was established with a 2D sketch on the reflection plane, a 3D sketch was then used to find the actual path BC, ending at point C on the surface of the first sphere. The 3D sketch contains a single line with one endpoint coincident with B and collinear with the direction vector just created. The final unconstrained endpoint of the 3D sketch is coincident with the surface of the first sphere, determining the point of intersection between the ray and the sphere.

In order to reflect the beam off the surface of the sphere it is first necessary to find the surface normal at the point of reflection. There are two ways of achieving this, both using a 3D sketch containing a single line. One end of the line must be coincident with point C, the endpoint of BC on the surface of the sphere. It can then be set to be normal to the surface of the sphere, at this point, in one of two ways. Its other endpoint could be constrained to the center of the sphere, or alternatively the line could be constrained to be perpendicular to the surface of the sphere.

The reflection was then simulated as for the previous reflection. The plane of reflection was fully defined as coincident with the surface normal vector and the point at the other end of the incoming ray, B. The direction of CD was then found using a 2D sketch on the reflection plane, with a line that is symmetric to the incoming ray, about a centerline on the surface normal vector. Finally, a 3D sketch containing a single line was used to find the actual path CD with the point D coincident with the surface of the second sphere.

The process was repeated for the remaining sections of the ray path. In general, the process has four steps:

1. Create a surface normal vector on the reflection surface, using a 3D sketch containing a single line.
2. Create a reflection plane which is coincident with the surface normal vector and a point at the other end of the incoming ray.
3. Create a direction vector for the outgoing ray, using a 2D sketch on the reflection plane with a centerline on the surface normal vector and a line, which is symmetrical to the incoming ray about the surface normal vector.
4. Find the point of intersection for the ray path, using a 3D sketch which is coincident with the direction vector and has an endpoint coincident with the geometry representing the surface of reflection.

The geometric constraint-based sketches in parametric CAD programs such as SolidWorks can be adapted for use in many ways. This tool is a powerful and intuitive way to solve many engineering problems, which often have nothing to do with modeling part geometry.

 

 

The votes are in and the winners of the 2020 LEAP Awards (Leadership in Engineering Achievement Program) were announced in a digital ceremony with products across 12 categories, including the category of software.

Critical to LEAP’s success is the involvement of the engineering community. No one at WTWH Media selected the winners. Instead, our editorial team did the arduous work of assembling a top-notch independent judging panel, comprised of a cross-section of OEM design engineers and academics — 14 professionals in total. This judging team was solely responsible for the final results.

The Bronze went to Mentor, a Siemens Business Unit, for its Observation Scan.

ICs for safety critical applications need in-system test to detect faults and monitor circuit aging. Scan-based Logic Built-In-Self-Test (LBIST) is the technique used for in-system test, but traditional LBIST often can’t meet the coverage goals within the required diagnostic test time interval (DTTI). A new LBIST technology called LBIST-OST (Logic BIST with Observation Scan Technology) dramatically improves the efficiency of in-system test.

Traditional LBIST poses two challenges: it often cannot meet the 90% coverage goal required for Automotive Safety Integrity Level (ASIL) D certification and fault detection happens only during the capture phase of the LBIST cycle. Each LBIST cycle has two phases—shift and capture. There are as many shift cycles as the length of the longest scan chain. The capture phase uses the values written into the scan chain to drive the combinational logic and capture potential faults. That means all the time spent on the shift phase is time faults are not being detected.

Tessent LBIST-OST employs control and observe test points with a new scan cell functionality that essentially connects the observe points in separate chains. This allows them to capture circuit responses not just during a capture cycle, but also during the shift cycles.

Tessent LBIST-OST delivers up to 10x faster in-system test than traditional LBIST and reduces the number of test patterns needed.

 

The Silver went to CADENAS, for 3Dfindit.com, a visual 3D search engine for manufacturer components.

3DfindIT helps engineers, architects, and designers quickly find, configure and download manufacturer-certified CAD and BIM models. With eight different ways to search millions of 3D CAD & BIM models, from 2300+ manufacturer catalogs, 3DfindIT transforms how designers’ source and specify components for their designs. Once the desired component is found, users can configure and download in more than 150 native and neutral CAD and BIM formats, ensuring the model works seamlessly with their specific deign application. Since all content is provided directly by the manufacturers, the user can be confident the data is accurate and up-to-date.

 

And the Gold went to Lattice Semiconductor for its Lattice Propel software.

Lattice Propel is software designed to accelerate development of applications based on low power, small form factor Lattice FPGAs. The design environment empowers developers of any skill level to quickly and easily design Lattice FPGA-based applications by enabling the easy assembly of components from a robust IP library that includes a RISC-V processor core and numerous peripherals. The Propel design environment automates application development for developers serving the communications, computing, industrial, automotive, and consumer markets.

Propel combines two tools, Lattice Propel Builder (for IP system integration) and Lattice Propel SDK (for application software development). An easy to use system IP integration environment, Propel Builder provides tools to integrate processors and peripheral IP. The graphical integration environment features an easy to use drag-and-drop, correct by construction methodology. All commands are Tcl scriptable. A seamless software development environment, Propel SDK is a software development kit (SDK) with an integrated industry standard IDE and toolchain. The SDK features SW/HW debugging capabilities along with software libraries and board support packages (BSP) for Propel Builder defined systems.

The judges commented: “Lattice structures create enormous potential in the area of design for additive manufacturing.” Congratulations!

Choosing an element type for a structural Finite Element Analysis

Dr. Jody Muelaner, Ph.D. CEng MIMechE

When performing structural Finite Element Analysis (FEA), it is often advised that thin-walled parts should only be meshed using solid elements if it is possible to use at least three elements through the thickness. When this is not practical, shell elements are advised. What usually isn’t explained is that this advice is based on the use of first-order elements, which are rarely used in modern FEA software. When second, or even higher, order polynomial elements are used, good accuracy can often be obtained using a single solid element through the wall thickness. What is often more important, is the number of elements approximating tightly radiused curved geometry. There are also many cases where shell elements can dangerously underestimate stress in key features. We’re going to use a number of simple parts and mesh convergence studies to illustrate these issues and provide the understanding required to select appropriate element types.

Element types
Before getting into the examples, let’s take a moment to review a few basics of FEA element types. FEA can be used to simulate a range of physics-based problems including heat transfer, fluid flow, and electromagnetics, but structural analysis is the most common application and this is what we will focus on.

Problems can be represented in either two or three dimensions. One example of a two-dimensional problem is a plate loaded in plane stress, perhaps with a stress raiser such as a hole. In such a case, differences in the stress through the thickness of the plate can be ignored and it doesn’t actually matter how thick the plate is, provided the load per unit thickness is correct. Another example of a two-dimensional problem might be an axisymmetric pressure vessel. We won’t consider one dimensional problems since the issues are very different to those of three dimensional problems. For most designers, running simulations directly from CAD models, a three-dimensional analysis will be carried out.

For a three-dimensional analysis, the elements themselves may be one-dimensional, two-dimensional or three-dimensional. Beams and bars are examples of one-dimensional elements, these elements are represented as a line. Although they may have a cross section associated with them, this is only used to determine their cross-sectional area and, in the case of beams, their second moments of area. Although the element itself is one-dimensional, it exists within a three-dimensional model meaning that the element can connect to other elements, or have forces acting on it, from the x, y or z direction.

The two types of elements considered in this article are shells and solids. Although a first order shell element is two dimensional, it can transmit bending forces as well as plane stress, allowing multiple shell elements to approximate three-dimensional structures, it should not be confused with the planar elements used in a two-dimensional analysis. Shell elements can be triangular or quadrilateral. Solid elements can be either tetrahedral or brick shaped.

Elements are represented mathematically as a polynomial and therefore have an order corresponding to the order of the polynomial. A first order triangular shell element has three nodes, one at each corner, and stress and strain can only be linearly interpolated between the nodes. A second order shell element has midpoint nodes on each side, giving a total of six nodes. Second order elements are also known as quadratic elements and allow stress and strain to be approximated using a quadratic function. A second order shell element does not have to be flat, its geometry can also approximate a curved surface using this quadratic function. It is also possible to use higher order polynomials to more accurately approximate curved geometry or changes in stress through the element.

Example 1: Flat rectangular plate loaded in bending
The first example used to compare the results from solid and shell elements is a simple rectangular plate loaded in bending. This simple model enables the plotting of simulation accuracy against mesh size. The rectangular plate is modelled with an elastic support at one end, a roller support at the other end and a distributed load over the entire upper surface. These boundary conditions avoid singularities and put the maximum stress in the middle of the plate, away from the supports. The image below shows the plate meshed with shell elements, one with a very course mesh and the other with a very fine mesh. The plate was flat in its unloaded condition and the curvature shown is a scaled representation of the deformation under load.

The simulation was run with second order triangular shell elements, with first order tetrahedrons and with second order tetrahedrons. These are referred to as simply shells and solids. A baseline simulation with five second order solid elements through the thickness was taken as the reference value. Each result was compared with this reference to see the percentage error in maximum von Mises stress and maximum deformation. These results are plotted against the mesh size, as a multiple of the plate thickness, in the charts below.

The results show that, when second order solid elements are used, a single solid element through the thickness is just as accurate as a shell element. Even solid elements which are considerably larger than the plate thickness give good results, in these cases there was a single element through the thickness but it had a larger aspect ratio so its plane dimensions were larger than the plate thickness. For solid elements which were three or more times the plate thickness the accuracy started to decline. Shell elements only give a significant advantage when the mesh is very course, with elements much larger than the plate thickness. First order solid elements are considerably less accurate, even with a very fine mesh, and for course meshes with only a single element through the thickness they give completely unreliable results.

This simple example seems to indicate that using higher order elements, as is standard with modern FEA software, it is not necessary to have multiple elements through the thickness. A 2-mm mesh will give very good results for a 1-mm wall thickness.

Example 2: Bent plate demonstrating element size relative to radius
The second example uses a rectangular plate which is bent in the middle with a radius at the bend. It was modelled with a symmetry plane. One end has an elastic support and the other end has a tensile force applied. Because the plate is bent this tensile loading attempts to straighten the plate resulting in a peak stress at the bend. Only second order solid (tetrahedral) elements were used in this example.

This scenario was simulated with a range of parameter values. The plate thickness was maintained consistently at 1-mm but the radius was simulated at values ranging between 1-mm and 40-mm. For each radius a reference simulation was run with four second order solid elements through the thickness and further refinement so that in the region of the bend, the elements were no more than 1/20th of the bend radius. Results were compared for global mesh sizes varying from 0.25 wall thickness (four cubic elements through the thickness) to three times the wall thickness.

The below chart shows the accuracy of each simulation plotted against the mesh size as a multiple of the wall thickness. It is clear that there is no correlation between mesh size and wall thickness. This remains true for elements up to three times larger than the wall thickness.

The next chart shows the accuracy of each simulation plotted against the mesh size as a multiple of the radius. In this case there is a clear trend. When the mesh size is 1/20th of the radius the errors were never more than 1%, however, when the elements were larger than the radius the errors were always more than 10%. For meshes that are 1/10th of the radius then errors are never more than 6% and typically much better than that.

Conclusions
The conventional advice for the meshing of thin walled structures is that shell elements should be used unless a solid mesh is able to achieve several elements through the wall thickness. With modern higher-order polynomial elements this advice is no longer relevant. A single solid element through the thickness will achieve results that are just as accurate as a shell element. This means that the considerable effort required to prepare geometry for shell meshing can be avoided. A more important consideration seems to be achieving a fine mesh to accurately model tightly curved sections of a thin walled structure, especially if high stress occurs at these areas. This indicates that curvature based meshing is a very useful tool for thin walled structures. Using higher-order solid elements together with a curvature based meshing algorithm, modern FEA software is able to achieve highly accurate results with very little pre-processing of geometry. However, caution must always be observed as there are many other ways that FEA can produce spurious results. Verification by physical testing remains highly advisable.