As I am contemplating hiring someone the coming year, a lot of thought has gone into what to expect from such a person. There are certain personality traits that need to be in place; a curiosity when it comes to the technical solutions, a willingness to listen, a systematic approach to problem solving using the formal systems that are already in place, and persistance/grit when it comes to being a stuck in a problem (you are guaranteed to get stuck with these types of tasks). Such a position would require life long learning, and that is not for everyone.
No matter the personality, I think they are some technical and mathematical aspects to simulation work at the highest level that are needed to have explored, and/or be interested in exploring in the position. Signal processing is large part, there is some general mathematics that will be used time and time again, and there are many topics in physics that go into the simulation setup. I have made a list (not exhaustive in any way) of questions that I could imagine would be used in an interview setting, and you can have look and see if this is something that you could imagine fiddling with every day.
For now, there is no position, but with the influx of tasks, both more traditional simulations tasks, but also training of entire teams and presentation work, I am at crossroad where I either turn down tasks or expand. So it would be good to already now get some conversations going.
The questions are not to be considered as make or break, but for getting a discussion going. Questions in red might be a little tricky... but readers of my blog will probably be helped somewhat.
Signal Processing
Assume transfer functions written in form of rational functions.
Consider the pole-zero diagram below, with no zeros (or zeros at infinity, if you like)
- Write an associated transfer function with general parameters
- Which order is it?
- What do you know about the value of Q (or damping factor, if that is your preference)?
- Which type of filter is this?
- Sketch the frequency response; amplitude and phase
- Is it minimum or non-minimum phase?
- Is it stable?
- What assumption did you make about causality for the above question?
- Assuming that it is a physical/causal system, sketch Region of Convergence. Explain.
- Sketch the pole-zero diagram for a first-order or second-order allpass filter.
- Is it minimum or non-minimum phase?
- Sketch impulse or step response. Explain.
Consider the pole-zero diagram below:
- How can this system be constructed?
- Explain positive vs negative frequencies
Consider the frequency response below.
- Which type of filter is this?
- Is it minimum or non-minimum phase?
- Sketch the output for any input of choice
Consider the output response below to a sinusoidal input and discuss (modal behavior)
- In your own words, describe the relationship between the Laplace transform and the z-transform
- Why do need consider special mappings such as Matched z-transformation, Impulse Invariant z-transformation, and Bilinear z-transformation?
- Are they conformal mappings?
- Explain aliasing as a phenomenon from any viewpoint you like (s to z-transformation, sampling, ...)
- Explain how Region of Convergence in continuous time (s-domain) relates do discrete time ROC (z-domain)
Mathematics
- Give your own overview of complex numbers
- Relate to polynomials
- Relate to phasors
- Explain eigenmodes from any perspective of choice (eigenmatrices and vectors, eigenmodes of physical systems)
- Relate eigenfunctions and eigenvalues to transfer functions (both from the aspect that the transfer function value itself can be viewed as an eigenvalue to certain eigenfunctions, and that the poles can be considered eigenvalues to the same eigenfunctions. May require some State Space Model knowledge.)
- Give an overview of Fouries Series and/or Fourier Transform
- Give an overview of Laplace Transform
- Relate Fourier and Laplace Transform to each other, and to signals vs systems. Why is Laplace Transform often considered one sided only?
- Relate the above to Region of Convergence by looking at the associated integrals.
- Describe Taylor expansion
- Describe Padé approximants
- Describe convolution (e.g. make a "discrete convolution machine")
- In any detail level you want, describe an optimization routine (conceptually, at least)
Acoustics
- Write the Helmholtz wave equation and describe possible solutions
- Walk me through the parts in the Helmholtz-Kirchhoff integral
- Describe the Rayleigh integral
- Discuss point multipoles (monopoles, dipoles, ...) and their characteristics (near-field, far-field)
- Relate to oscillating and 'breathing' sphere solutions
- Explain room modes
- Explain how placement of loudspeakers will affect the resulting pressure near modal frequencies
- Explain the equivalent of voltage and current in acoustics.
- Write an expression for acoustical impedance and check units
- Describe the sound field from a piston in an infinite baffle?
- How does the phase of displacement related to phase of pressure in this situation?
- Why does a loudspeaker eventually roll off at higher frequencies? Explain all effects you can think of.
- For a two-way loudspeaker consisting of minimum phase section for the high frequencies (tweeter and its cross-over section) and a minimum phase section for the low frequencies (woofer and its cross-over section) will the total response generally be minimum or non-minimum phase?
Structural Mechanics
- Describe analytical solutions to examples of your choice (bending beam, membrane excursion, ...)
- Explain the equivalent of voltage and current in structural mechanics.
- Write an expression for mechanical impedance and check units
- Describe non-linear geometry and where it is relevant
- Describe non-linear material parameters and where it is relevant
- Describe non-linear contacts and where it is relevant
- Describe characteristics of a mass-spring system (frequency response, modal behavior, DOFs, order)
- Explain eigenmodes with examples and the effect of boundary conditions
- What happens when a source is applied to the above examples?
CAD Specifics
- Walk me through different boolean operations
- Describe how to handle overlaps and gaps
- Describe assembly vs part
- Describe mates
- Describe constraints
- Discuss export options
Simulation Specifics
Consider the sinusoidal force input below in a steady-state setup:
- Sketch the associated signal over a few periods
- What would it take to create a real sine signal input?
- And for cosine?
- Relate this to positive and negative frequencies
- How would you run the simulation for a real input?
- Describe symmetry conditions for acoustics and structural mechanics, respectively. Why is there a difference?
- Describe the coupling between acoustics and structural mechanics
- Describe element order in your own words (sub-, iso-, superparametric)
- For a given order of acoustic pressure, what is the associated order of the particle velocity?
- Explain Principal stresses and their axes
- Describe differences between BEM and FEM (dimensionality, implementation)
- How do we do external acoustics problems with FEM? Are they external in the same sense as a BEM analysis?
- Describe possible numerical issues with BEM
- Ditto but for FEM
- Which material parameters must be known?
- Explain non-linear geometry effects with examples, and how to handle them simulation-wise
- Explain non-linear material effects
- Discuss harmonic/steady-state vs transient solutions
- Which effects are not included in the former for acoustics and for structural mechanics?
- What happens if you check the Include Non-linear Effects box in a harmonic analysis?
- How does the real part and imaginary part of a result relate to phasors? Since we make measurements in the real world, is it enough to only consider the real part?
- How do simulated modes compare to textbook modes? (Orientation, degenerate modes, ...)
- Describe Shape and Topology optimization
Concluding thoughts
As already mentioned, the above is to illustrate some of what a position at Acculution ApS would entail, and at the very least the questions can serve as a test for engineering departments to gauge where they are at. If all this is right up your alley, then let us connect and see where it takes us.