  To elucidate the key factor for the quantitative prediction of the
shear-thickening in suspensions in viscoelastic fluids, direct numerical
simulations of many-particle suspensions in a multi-mode Oldroyd-B fluid are
performed using the smoothed profile method. Suspension flow under simple shear
flow is solved under periodic boundary conditions by using Lees--Edwards
boundary conditions for particle dynamics and a time-dependent oblique
coordinate system that evolves with mean shear flow for fluid dynamics.
Semi-dilute many-particle suspensions up to a particle volume fraction of 0.1
are investigated. The presented numerical results regarding the bulk
rheological properties of the shear-thickening behavior agree quantitatively
with recent experimental results of semi-dilute suspensions in a Boger fluid.
The presented result clarifies that an accurate estimation of the first normal
stress difference of the matrix in the shear-rate range where the
shear-thickening starts to occur is crucial for the quantitative prediction of
the suspension shear-thickening in a Boger fluid matrix at around the
Weissenberg number $\rm{Wi}=1$ by an Oldroyd-B model. Additionally, the effect
of suspension microstructures on the suspension viscosity is examined. The
paper concludes with a discussion on how the flow pattern and the elastic
stress development change with the volume fraction and Weissenberg number.
