The technical textiles company, which has its roots in Dundee and still employs around 120 in its yarns division there, said Australian Steve Good will be its new boss. Mr Simpson was most recently chief executive at Belgian company LBC Tank Terminals Group.
Before that he spent more than two decades with The Dow Chemical Company after joining it as a production engineer.
He will join Low & Bonar on August 26 and replace Steve Good as group chief executive from September 8.
Mr Good, who has been with the business for more than a decade and chief executive since 2009, plans to pursue a portfolio career of non-executive roles.
Martin Flower, chairman of Low & Bonar, said: "[Brett's] extensive international experience in general management, marketing and commercial and production roles in the chemical and plastics sector will be of great benefit to Low & Bonar as we embark on the next stage of our development to establish ourselves as a global performance materials business."
Separately, the company indicated its current trading has been strong across most of its sectors. Sales of civil engineering and building products were ahead of relatively weak prior-year comparisons while there was also strong demand from the industrial and leisure sectors. Demand was said to be muted in the flooring and automotive markets.
Low & Bonar said it had good trading momentum going into its traditionally stronger second half of the year. Shares closed up 1.75p, or two per cent at 82p.